Which version of bootstrap? I see there is a QCCR (QCCR1J83339) out from HP that is in the investigation/under review phase for a request on bootstrap 3.3.5 support being added to Sprinter 12.50.
I have seen what you point out with some jQuery applications, where the drop down can be navigated but not choice can be picked during a Sprinter record and mirroring action. Would seem then to be a limitation on what Sprinter supports for the web technology in this case.
Good the QCCR already exists. Hopefully integration of the technology will be forthcoming.
For now, you can catch up on your mirrored machines manually for dropdown choices.