Thats great...

Its doubtful that a filter will correct the no sync issue..It would help remove any noise on the lines though that would show up in the colours. But unless you have an issue like that i wouldn't bother..

If you want to keep the amp you could use it for the RGB signals and simply link the sync signal past the amp..

Or the chassis you have might just have trouble with a composite sync signal even though it does support it. So you could cut the sync trace ( should be labeled CUT FOR H V SYNC or similar) on the amp PCB and feed out separate sync signals. Your chassis might like that better.. This is assuming your amp PCB is not faulty..

Honestly the AMP PCB is not needed,  a correctly adjusted chassis will produce a better result every time. The only time one of the amp PCB's would come in useful is if the chassis you were using had very limited RGB adjustments..