I think I have a handle on what is going on here. When you enter Accurate Play Solo, you are automatically moved to the Client context but the open script still belongs to the Edit context. Breakpoints in stock Studio are restricted to the context they are set in so you are unable to change the Edit context breakpoint from the Client context. If you open the Client version of the script (indicated by a blue dot on the tab), you will be able to disable the breakpoint.
This is annoying! This is why we released Breakpoint Replication a few months ago: Universal Breakpoints Beta - One Breakpoint to Rule Them All. With this Beta feature turned on, you won’t have to worry about context anymore and can edit breakpoints from anywhere.
(There is still a small bug here. With ‘Breakpoint Replication’ beta turned on, you can disable breakpoints in Edit context but the breakpoint icon doesn’t update (it should - we’ve ticketed for a fix). However, you can view the true breakpoint state from the Breakpoints widget.)