If you want to make stuff with the Traveller name, you have to sublicense from Mongoose. If you want to use the T20 text and material, you need to talk to Hunter as well. If you want to use the Pathfinder material as the core, you need to talk to Paizo.
Paizo has an open license, as does Mongoose, but those two might not be flexible enough for what you are looking to do. Mongoose also allows paid licensees who have other freedoms; any Mongoose license will sunset on or before their sunset, or if they lose their license for some unforseen reason. (This isn't a knock on Mongoose; it is a real but slim possibility with most licenses, including Hunter's, that premature termination can occur for any of a variety of reasons.)
Note that the licenses are separate from the text; Paizo has an SRD for PFFRPG. Hunter does not; Hunter does have Marc Miller's permission to "detraveller and reuse" the T20 ruleset.
Best advice, even if using just the free licenses: hire some time with an IP attourney in your local area.
Note that mongoose has TWO separate overlapping licenses you need to deal with if not buying a paid license: The Wizards OGL 1.0a, and the Traveller Trademark License. If you don't adhere to the OGL, you can't use the SRD text. If you don't adhere to the TTL, you can't use the distinctive elements of the Traveller trademarks unless you negotiate a separate license with Mongoose.
Part of Mongoose's license is exclusivity; only SJG is permitted to renew direct with MWM for English RPG books. (Whether they will or not is yet to be seen.)
