Public selves

People can tell which strategy you are using, and usually the things that are ‘bad to show’ are bad for you to show, but other people would be perfectly interested to see them. So it is less cooperative, and people may respond to that, which may on a longer term view be bad for you.

Related to this: if you are hiding things, then people may detect that you are hiding them, but not what you are hiding. As a result, they can't tell whether you are hiding something relatively innocuous or something worse, and may intuitively trust you less. If your strategy tends towards revealing things, then they might see the bad things but be overall convinced that you're not hiding anything that would be even worse.

Some people will likely also find your bad sides relatable.

At the same time, sharing too much too quickly can also send a negative signal, as it may suggest a lack of discernment about social norms. 

What seems like a principled intermediate option is to find or build a 'Bubble', i.e. a social environment in which one feels (and ideally is) safe to reveal more of one's self publicly.

A few considerations:

1. The primitives here aren't exactly the ones I would use, for instance "oneself" is making an assumption about the nature of self that seems to extend to the rest of this essay, but it's something that my felt sense of has changed over time to where it's no longer a fully natural category. There's no longer "oneself" that I can put a mask on, or that I can display a full version of.
2. The assumption of an optimal strategy for all situations seems also quite off to me.  The way I relate to how "cognitive/filtered I am vs. "present/spontaneous" is moderated by several specifics of any given environment, situation, social group, or intention.  I may be more filtered in an environment that's unfamiliar and has a lot of unstated expectations, I may be less filtered in an environment where there's not a specific intention I'm holding.  I would think much more about "When should I be more and less filtered" than "Should I be more or less filtered."
3. This idea of shaping your identity doesn't have to feel like hiding or stifling.  It can feel much more like playing, going to a costume party, or singing. It can be like finding a new range, or trying a new singing style. Playing with your range of expression can be quite fun.

The notion of weirdness points has never spoken to me, personally, because it seems to collapse a lot of social nuance into a singular dichotomy of weird/not weird, and furthermore that weirdness is in some sense measurable and fungible. Neither, I think, is true, and the framework ought to be dissolved. So what's goes into a "weirdness point"?

• How familiar is the idea? - Vegetarians/vegans are a little weird, but most people probably know a handful and most have a notion that those people care about animal welfare and maybe some even know about nutritional ideas or the effect of meat on the climate. Cryonicists are extremely uncommon and their philosophy is not widely spread so people need to do a bigger intellectual lift to understand them.
• How appropriate is the sharing? - A vegan has an understandable reason to mention their diet almost any time a meal is shared, but if they never stop talking about it at parties, people will be annoyed and less sympathetic. The appropriate time to bring up cryonics is... During discussions about philosophy of death? Futurism? Maybe you can get away with it if someone just asks you what you're reading lately.
• How demanding is the idea? - People tend to not be huge fans of being asked to do things they wouldn't normally want to do. This is of course a fundamental obstacle to anyone looking to change the world for the better, but it still bears consideration. More demanding ideas require more compelling evidence and more time to allow people to come around to them, and will often require a lighter touch up front to not be dismissed entirely.

All of these factors and more besides will constitute the weirdness of an idea, but to me none of them suggests the best strategy is to hide your ideas. It seems to me that dissolving a weirdness point just tells us something we could probably have figured out in the first place— weirdness exists only in social contexts and can thus be moderated by just developing better social skills. I can be honest about the vast majority of beliefs I hold by just picking the right moments to share them and choosing the way I frame them based on my understanding of the points above. That's not propaganda papering over a forgettable version of myself, it's just correct gameplay.

Epistemic status: largely just brainstorming 

I think there are a number of examples in history of people or groups of people successfully advancing the cause of ideas significantly outside the societal mainstream through living and embodying those ideas fully openly and authentically. One example that comes to mind is FM-2030, who was pretty much the exact opposite of choosing to spend one's weirdness points selectively and did a lot to spread transhumanist ideas as a result. Anther good example would be the post-Stonewall gay liberation movement. 

Also consider whether you have to right to be forgotten - Depending on the medium through which you spend a weirdness point, what might seem like a fair if not uncommon point of pride and character could in the future come back to bite you. 

Your description of "cowering lonely behind a cardboard cutout of the most forgettable person while proffering optimized propaganda through carefully selected slots" seems like behavior that would be characteristic of somebody whose every word and action is captured and potentially retrievable for an indefinite amount of time.

One size does not fit all.  "the public" is not homogenous, and individuals have very different ability and tolerance for crafting their presentation to different audiences, and different audiences react differently to various dimensions of presentation.  This is almost unrelated to how "true" the presentation is.

I don't think there's any way to avoid some amount of strategizing.  Most people do it fairly unconsciously, and until very recently in human history it wasn't commonly acceptable to even ask the question.  Arguably, this question is the primary driver of human brain evolution.

Your weird trait X may make (better) sense in presence of weird trait Y. If you show X and Y, you show a possible alternative. If you show X and hide Y, you show something that doesn't make sense.

(I don't have any convenient example in mind, just a vague feeling that something like this might be possible.)

Making correct decisions is hard. Sharing more data tends to make them easier. Whether you'll thrive or fail in a job may well depend on parts you are inclined to hide. Also, though we may be unwilling to change many parts of ourselves, other times we are getting in our own way, and it can help to have more eyes on the part of the territory that's you

I think this post is conflating two categories of facts about myself that I might want to conceal: the weird and the disreputable. 

Example: if I only listen to 18th-century Japanese music, that's weird but not disreputable. If I'm addicted to phenibut, that's weird and disreputable. (There are also "normal" facts that are disreputable, such as "I'm depressed.") OP's example of "personal problems" would fall into the disreputable and potentially weird category.

A person's answer to the "should I be openly weird" question might not dictate their answer to the "should I be open about disreputable things" question. It can be quite dangerous to be open about disreputable things about yourself; those facts could end up destroying your relationships, ending business opportunities or getting you fired or arrested. Yes, ideally we would be in environments where we could trust our friends to keep private information private, but, people being people, you should not expect ever to be in such an ideal environment. Don't tell a friend about your personal problems unless you have accepted that everyone in that group will come to know them.