Respond to the interview invitation late
I can't imagine you would be knocked out of the running because of your delayed response.  However, since this will create something of a headache for the scheduler, I hope you can offer them a lot of flexibility in the scheduling.  For example, if you have any non-urgent doctor's appointments coming up (during working hours in the target time zone) in the next two weeks -- if possible, reschedule them.

Make sure you outline very clearly any remaining constraints on your interview availability.