The more I've looked into it, the more I realize only a few integrated amps have BOTH the home theatre bypass AND the trigger input features I mentioned (Marantz does in your price range but only a handful of others)...worst case scenario without the trigger input is you're turning on one extra box (probably via remote and maybe from the same universal), so don't let the trigger input be your make or break feature. Missing out on a home theater bypass setting is a bigger headache for this type of combo setup because you have to figure out a reference volume for the IA and change the volume on both devices when you switch sources. That list I posted earlier isn't all-inclusive (I know the Nuprime IDA-16 has HT bypass and isn't on the list), but it will give you more than enough good options to consider and a decent, if not comprehensive, reference if you find something you're interested in but aren't sure if it has HT bypass buried somewhere in it's menu options.
Good luck and happy listening!
Good luck and happy listening!