These are intermediate steps that have helped, giving us some time to figure we whether want to migrate to a different database:
1) We started using InnoDB a few years ago.
2) One of our common queries joined parts of multiple large tables; we essentially cached the useful parts of the join and put the results into a memory table.
3) Our non-production server is under a lot of load from our back-end processes, and we decided to move the database to a RAM disk, which greatly improved performance. It's important to note that, while inconvenient, a power outage is not catastrophic in our case (e.g. we don't deal with financial transactions); we do daily backups to non-volatile memory. For around $5,000, you can build your own server that has over 250 GB of RAM and then put it into a co-location facility.
We seriously considered migrating to Postgres, but #3 has bought us some time on that front. It might be that we eventually still migrate, but we prefer to do it after the company is on better financial footing. It might also be that while we wait, other less painful options open up over time (e.g. maybe MariaDB will suffice).
I hope the situation resolves well.