I know/think that ~ばいい can be used, ex.
How am I supposed to deal with this?
But is there another way to say it? Maybe something involving はず?
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Well, I still don't really understand the exact situation you're imagining, but maybe something like
is desperate enough for you? (At least, I can imagine someone crying and saying it...)
The closest phrase to "be supposed to do" is …することになっている, and you can say どう扱うことになっている(のだろうか)？. However, if the matter doesn't have enough reproducibility, it's not necessarily a useful expression. In that sense, どう扱えばいい or どう扱うべき are reasonable after all.
Your example (どう扱えばいい？) is about as close to "How am I supposed to deal with this?" as you can get.
Here are some related phrases:
What should I do?
What is the best way to (do something)?
There's no way I could do that.