From: Winnipeg, MB
I use xAPs for amphib invasions all the time in '42 and early '43 (on the Allied side). There's not much choice beyond a few APs. The trick is having a low unit density on each xAP or xAK.
I recently did some experimenting with landing radars at a newly taken base. Once I took the base and built the port to Level 1 (from 0) I was able to unload the radar from an amphib TF, even without naval support present (which surprised me, actually).
Regarding the original post, one thing that can alleviate that problem is having amphib TFs that only contain supplies. Once the supplies are unloaded, that's all the more capacity you have to re-embark your troops.
One final note: Years ago someone on this forum said that for ports Level 4 and up, transport TFs are faster to unload, while amphib TFs are better when unloading at Level 3 or lower ports. I haven't tested it, but that rule of thumb has worked well for me.
If the base just captured has a nearly undamaged port, I will unload the light stuff amphibiously and then dock the amphib ships to unload heavy stuff like tanks, engineering vehicles and larger artillery pieces. I don't like losing these in the water just because someone didn't tie the sling rope properly during amphib unloading!
No matter how bad a situation is, you can always make it worse. - Chris Hadfield : An Astronaut's Guide To Life On Earth