It's not really about Idul Fitri, Gili T is very busy in peak season, which has spilled over into September. You really need to make a booking unless you are prepared to be turned away.
Any impact on accommodation from domestic tourists will be in the lower end hotels usually, so turning up on a hope is not a good plan.
If you arrive Mataram 15,30 you will not have time to catch a public ferry to the Gilis. You may not even be able to charter by the time you get to the harbour.
Best plan would be to stay in Senggigi for the night, and then get the Perama bot to Gili T the next morning, which is a little more than the Bangsal ferry, but still reasonable, and more convenient than running the Bangsal gauntlet. By the time you factor in transport to Bangsal it is not much more. Also the trip down the coast is really nice.