The stability problem is quite fixable, but it'll take additional hardware. If/when you decide to give it a try, it would be helpful for you to join the forums and mail lists where the EME guys pass along information and find out just what the stability requirements are, and what they're doing about it.
EME is one project that you don't do by yourself in a vacuum. RR isn't the venue for getting the details, as there are EME mail lists that are dedicated to the topic. Keep in mind that these guys aren't beginners. You'd do very well for yourself by playing terrestrial weak signal for a while before you go on to EME.
You COULD probably trade your HF rig for something that has 2m SSB 100 watt capability
You're not going to find a decent transverter for $100. The Downeast 28-144 transverter is closer to $600 base price. Add a few hundred more for the synthesized LO for stability, supply your own 10 MHz reference oscillator - I prefer rubidium. There's a few hundred bucks right there. You could use a high quality TCXO instead for an order of magnitude worse stability than rubi, and that alone could cost you $100 - just for the 10 MHz stable reference source.
You could easily spend well over $1k just for an EME grade transverter.
You could consider trading up your HF rig for one that does 2m SSB at 100 watts. Cost would be the sasme or more than the add on transverter, but modern rigs are pretty good, stability wise.
If the interest is there, what you should do is start by building simpler things for lower frequencies and some station accessories. Get your skills up, and then tackle VHF/UHF transverters and weak signal modes.
Then, at some point, EME will be a natural progression and it won't seem like such a leap, like it does now.
