Shadow isn’t in Sonic Underground Remixed for a number of reasons, both meta and in-universe, but I’ve been thinking if he were in SUR, how would I tackle him? Think of this like an au within an au!

Shadow is firmly on the side of wanting to defeat Robotnik, but his prefers to work alone and do things his way so he refuses to join the resistance or any other organised group. He’s not ‘the ultimate lifeform’ in this universe, but rather a regular teenager (albeit with superpowers)
He helps the triplets get out of tight spots and tricky situations, and in particular develops a good-natured rivalry with Sonic

And just to state this again, very clearly: Shadow will not be in Remixed. This is only a ‘what-if’ situation for fun