You might be just fine adding an unindexed tsvector column, since you've already filtered down the results.

The GIN indexes for FTS don't really work in conjunction with other indices, which is why https://github.com/postgrespro/rum exists. Luckily, it sounds like you can use your existing indices to filter and let postgres scan for matches on the tsvector. The GIN tsvector indices are quite expensive to build, so don't add one if postgres can't make use of it!