connecting a GSM booster directly to a mobile device?
I don't think this is a good idea.

It might damage your phone to connect it directly. The repeater will transmit a significant signal indoors, to travel to your phone.

It will almost certainly damage the repeater to do this. Your phone produces up to 1 Watt of RF power, the repeater is expecting a much smaller signal.

But most importantly, I don't think a repeater will help with your signal problem. If the signal is too weak when directly connected to the phone, there's no reason to think their amplifiers will be able to do better.

On the receiving side, your phone will probably have a lower noise figure than the repeater - it needs to work through some filters your phone doesn't need. On the transmit side, the power is probably limited by license conditions, to the same as the phone's power.

Repeaters are for convenience, to free you from the direct cable. They do "boost your signal" compared to what would have leaked into the building, but that's not your question. See This Answer for a bit more about the signal levels.

You're already doing the best thing using an external antenna. Can it be improved? A higher gain antenna might help. Can you replace the coax with a much lower loss cable? These would help.