Don't see any problem with it personally. Some issues that might arise are excess salt from the algae, however I'd guess you'd need to compost a ton (literally) of it to see any issues. Others might be accidental reintroduction of non-native algaes due to birds,wind, etc.
That said, when I trim I toss a handful in planter box outside
I comport both types you mentioned and bury it in the pile so it does not have a chance to move away as a viable plant. I have put two basket ball sized balls of chaeto in the compost.