I know the owner of Scallywags (Trawangan and Air) and when he and his wife go there they use the Bluewater Express from Benoa. I believe Bluewater also offer a pick-up service in the Bali tourist area. There are many other boats leave from Padang Bai as it is closer.
Scallywags also offers a helicopter service but it's pricey...still, if you are generous that would be a wonderful present as it includes a trip around the coast via Mt Agung.
Air Bali - Bali Helicopter Charter - Official Website
The new airport in Lombok is far away from Senggigi..... it is a tortuous journey to get to the ferries to the Gilis.
I suggest you pick-up the latest bi-monthly Lombok Guide..it should be available in all Bluebird taxis and many hotels/bars around Bali. If cannot find I have one in front of me so ask questions and I'll try answers.