No need to pay the entire fee, if this really is a burning desire of yours, you can send any sized UTXO and then start to follow the trail, but there's no guarantee that watching this address will result in data points you can tie to a person though and it may take years for them to slip up.

I would rather you look at the previous addresses you've already interacted with and start your sleuthing there since they might have been more careless in how they transacted knowing they had you in the scam already.

If you can I would try to get addresses of others that have been scammed, the larger the data set the more chance you have of finding patterns and if one of those addresses interact with an exchange account, then you have a possible lead

Having been hacked on an exchange in the past, i went through this myself thinking it would help eventually I got to some rando exchange in Ukraine and they just weren't compliant so I just cut my losses