Since I'm still thinking of it, here's the last major component to the mix that made BattleTech: Crusher Joe.
In terms of BattleMechs, you really only get the Locust, so where the Crushers really come in is with the AeroTech and ground vehicle supplementary rules for the game. The primary Lance-scale Dropship is the Joe's starship. The two fighters it carries, and those fighters they engage in the movie, are all in AeroTech. Going through the early Technical Manuals for the game will reveal other Crusher Joe design, lifted entirely from the OVA and film.
The light novels and their film/OVA adaptations are not as hard as the BattleTech adaptations, but they're close enough that they do fit; lifting the entire "Crusher" concept would work for a Star League era game, before everything fell apart into the First Succession War, as Crushers are meant to be independent troubleshooter teams- and thus perfect premises for your common group of tabletop gamers to use for role-playing. For you MechWarrior/A Time of War people, this here is very much your huckleberry.
But again, all of this was likely unknown to the folks at FASA (or ill-known at best) because, like Fang of the Sun Dougram, this wasn't well-known in the West at the time and still isn't now. What Western releases we got were few, far between, and not that good. Finding legal sources is difficult, but possible, and (as with Dougram) worth watching so you can add to your imagination palette for Real Robot fiction or gaming. The fan-run resources are there, however:
Yes, the film's out there too. You know how to search, so have at it.
The original light novels, however, never got to the West. It's unlikely to get one anytime soon due to a new addition to the series, or a recreation, in the works or a rising Western fanbase ready to throw money at them. (The former is what drove those new English translations of the Legend of the Galactic Heroes light novels.) The other merchandise is likely hard to find, as that rarely made it across the Pacific Ocean. However, all you need to do is run an Image search to find official art work and stills from the media adaptations in bright, living color and compare them against their BattleTech incarnations to see the connection.
That about wraps up the anime origins of BattleTech. I may have missed an item or two of note (the sphereoid DropShips being the ones nagging me right now, and some other AeroSpace fighter designs), but between Fang of the Sun Dougram, Super-Dimensional Fortress Macross, and Crusher Joe that covers almost all of that conceptual ground. Enjoy, and remember: No Guts, No Galaxy!