How long do I actually have to bring a claim like this?

Your deadline depends on whether you had a written contract. With a signed written home improvement contract, Code of Civil Procedure section 337 gives you four years from the breach to file suit. If your arrangement was oral rather than written, section 339 shortens that to two years instead.

A license bond claim operates on a separate and generally shorter timeline of its own, tied specifically to when the bond was actually in force covering the work in question, rather than running parallel to your lawsuit deadline. Because of this separate and often tighter window, it's worth filing your bond claim sooner rather than trying to time it to match whichever deadline applies to your underlying lawsuit, waiting could mean the bond claim window closes even though you'd still technically have time to sue.