Was joining the local Ottoman harem seen as an acceptable way for a lower-class girl to achieve social mobility?

First off, we need to clarify what a harem was. There's a perception in Western culture that a harem meant a large group of wives/concubines/sex slaves for the "use" of Ottoman/other Middle Eastern nobility. This is incorrect. The harem was the women's quarters, forbidden to the outsider, yes, but it might be more accurately be thought of as the household. Much as European society eventually developed a split between the "men's sphere" of politics and social life and the "women's sphere" of the home and household, in parts of the Islamic world (the example I know best is Morocco, but the Ottoman Empire applies, as well), the harem signified the sacred and forbidden women's sphere. Yes, the household was the place where sex could propitiously occur, but that was (or at some point became) equally true in Europe - sex and family life were private.

Viewed from this perspective, the question you're asking essentially translates to, "was it acceptable for a lower-class girl to join a wealthier Ottoman household to achieve social mobility"? I think that the answer to that question would generally have been yes. However, I can't provide a definitive generalization, since such social considerations would have been different depending on which part of the Ottoman Empire you were in and what century it was - the Ottoman Empire was large and culturally heterogeneous, so circumstances in Ottoman Egypt would have been different from those in Ottoman-controlled Transylvania, for instance.