No-IF done other way, via the sneeky bypass of oebb.at, it is 3€ for the IC-these tend to run about ev 2 hrs along that adriatic coast-and on to Milano most. FRecce trains cost 10- the 2€ extra is a fee for EUrail-app. You can also do that for these prices at the local counter. Chance that you will be told ´all completo´ is near zil/nil. Plus you can in theory even go-with more changes, along by using local=regionale treni. EXCept when that SUN is the last day of an IT holiday (I dk when these are).
Best strategy for most is to book a later train and if arr ontime/early, go to desk and try for an exchange (these should be possible in theory at least). Fro memory: Bari is not an attractive town and the ferry also dock far away from centre/train