Can I travel from Labuan Bajo to Labuan Lombok by ferry with a motorbike?

You have two questions, here, actually.

The first is

Can I go from Labuan Bajo to Labuan Lombok by ferry

and it's mostly answered here.

The second is

Can I carry a motorbike with me

and this is a little bit trickier to answer.

Basically:

1. If it's a passengers-only ferry (there are), you can't load a motorbike
2. If it's a mixed ferry, as much as you can load a car on it you can always load a motorbike too
3. If it's a rented bike, you can't move the bike from province to province because rented motorbikes insurances cover you only in the province where you rented the bike.

More in detail: when buying the ticket for the ferry you'll be asked for the bikes document, and they will catch you in a moment. You can always bribe the officer, in Indonesia is doable, but it's a practice that is slowly getting more difficult and less widespread, especially with foreigners. You may bribe the ferry officer at the ticket booth, but after that you'll be forced to bribe every other officer that you'll meet during your trip and hope it works every time.

So, unless you buy a bike yourself and get the right insurance, as far as I know there is only a single company in Indonesia that rent bikes to tourists with full insurance coverage: Indocampers

Notice: I'm not in any way affiliated with them. I happen to know the company 'cause I spent a hell lot of time searching for a bike for my trip across Indonesia, and at the end of the day they are (were?) the only viable option.

One BIG warning: their bikes are all east asian models, and that means that they are all the size of a toy bike. If you, like me, are at least decently tall for a male, you'll find them really uncomfortable: I'd suggest you to totally avoid them if you are more than 180cm tall. And the suspension are on par, so again I'd avoid to do it if your total weight is more than 80kgs.