You mentioned the LIRR train out of the Atlantic Ave terminal near the Q. Note also that the East NY station stop on that same line is right next to the Atlantic Ave stop on the L (several miles down Atlantic Ave from the other station). This looks like it may be slightly faster than the other LIRR options all considered (but I haven't actually timed any of these and so I am just guessing).

According to the LIRR schedule, there is a train scheduled 6:38 East NY to 6:46 Jamaica (and another one 9 minutes earlier). The schedule for the L has it taking 27 minutes from Union Square to Atlantic Ave, and headways are 4-6 minutes. So it is possible that you might make it that way.

Obviously East NY is not the greatest of neighborhoods but I am assuming the transfer will be OK during daylight hours.

This was a fun game, but I hope you can successfully beg your client for the extra 15 minutes.