Starting with the basics. Backups do not take locks, so the timeouts can't be from there.

Before looking for a solution, I'd suggest identify the problem. Why do queries time out? It won't be due to locks, so what is making them take longer? What wait types do you see during a backup and in at what % of overall? What wait types do you see normally (when backups aren't running) and at what % overall?

With that info, you should be able to start targeting the specifics of the problem and planning ways to mitigate those specifics.