In many cases, however, it is assumed that a suitable transformer will
be selected by Lp, Isat and coil ratios. Is this acceptable
Yes, it's totally acceptable providing you adopt the approach of winding your own coil, with the core of your choice that is operated at the frequency of your choice within the design limits imposed by variations in the voltages on the incoming DC supply.
If you expect to find a suitable off-the-shelf transformer, more often than not you will be disappointed and so you are faced of biting the bullet and rolling your own. So, you roll your own and bravely connect it to your flyback circuit and, hey presto it works.
You are then left with that wonderful thought - "what was it that originally concerned me about making my own transformer".
Sure things can go wrong but these are not difficult circuits to debug and you'll probably never make that mistake again (such as not using the right dot notation for the secondary winding or miscalculating primary inductance). Anthing else is benign and sure, things may get a litle hot but you turn the supply off and double check things and rewind/re-core.
It seems to me all those calculations are supposed to be done already
by the transformer manufacturer.
And if that manufaturer is you, that will happen.
Regards the product recommendation, you can usually re-wind secondaries how you want - see - you are starting to get there!!