How to accommodate a range of input voltages
I don't know about the TPS55165 specifically, but buck-boost controllers are specifically designed to smoothly transition from the input being lower (or about equal to) the output to the input being higher.  One would hope they'd do the job with ease, but then, it's not the easiest job.

Scrutinize the data sheet, while reminding yourself that it was written to (A) sell lots of chips chips, while (B) keeping TI from getting sued.  So if they say something definitive, you can trust it (see B), but if they only seem to say something definitive, be suspicious (see A).  And note that I'm not picking on TI -- all semiconductor datasheets work pretty much the same way.

They do have a "step down to step up transition" graph in there that looks pretty good to me, but check for yourself.  I am, after all, just some guy on the Internet.