Not vacuuming should not be considered an option. Only analyzing just updates the statistics and does not do necessary maintenance functions on the table. The longer you let this table go unvacuumed, the worse it's going to be when it does actually need to be vacuumed. At some point it will be required for it to be vacuumed. It will cause autovacuum to kick in a more aggressive vacuum first (look into autovacuum_freeze_max_age) or your database will shut down (look into transaction ID wraparound). The other more immediate issue is that queries will start to become slower and slower as old rows are not being marked as reusable and bloat continues to grow.
If this table hasn't been vacuumed for a very long time, a good chunk of the size could indeed be due to bloat. I highly recommend using pgstattuple or the tool discussed in this blog post to check
One thing you could do would be to do the ANALYZE separate from the VACUUM. That will at least shorten the transaction time since it doesn't have to wait until the ANALYZE finishes to commit. But then you should do the ANALYZE soon after the VACUUM finishes.