A visor swap will definitely get the job done. Don't forget that the Marks with/without the sunroof utilized different visors. The HomeLink transmitter resides entirely within the visor.
You could also grab one out of another car and just swap the module. the visor opens like a "clamshell," and the module taps into the vanity light for power. The one in my '99 Town Car for example, would be identical, and easier to find in a junkyard.
Just note that not all "HomeLink" garage door openers are compatible with the transmitter that came with these cars. The two Chamberlain openers I installed when I bought my house in 2009 for example, are. The new Chamberlain opener I just added last year, is not, and required a plug-in transceiver that listens on the old frequency and rebroadcasts on the new one. I've only managed it to work well on 1 out of three Lincolns in my fleet. (so far)