Note that some repeaters have a "kerchunk filter" that eliminates the hangtime (aka "tail") if you only transmit briefly. Assuming it normally has a hangtime, are you keyed up and saying something for at least 10 seconds? The brief green light on the mobile would be consistent with no hang time on the repeater, or a kerchunk filter.
Are you able to talk through the repeater, using the handheld, to another user? Does the mobile hear the other user (through the repeater)? No RX/decode CTCSS or DCTCSS or "optional signalling" on either, right?