As a fellow Lefty, I can tell you... if you have the opportunity to buy a nice "Left bolt" rifle; DO NOT pass it up. Buy it.
At the same time, there would be nothing wrong (at all) with building a varminter on a large frame martini action. Maybe in something rimmed like a .225 Win. There is also the .222 Rimmed, but brass is a bit spendy.
Then there is, of course, inventing your OWN cartridge.. take a rimmed case that is plentiful and cheap ( .303 brit, or .30-30 come to mind), saw it off short, neck it down to .22 or .25.. and there you go. (of course, I'm oversimplifying just a tiny bit...).
The 40XB is a spendy piece of hardware if you're looking at new... but if you've located one used, quit yoru browser, grab your Visa, and go buy it! NOW! They're VERY uncommon in left hand, and will make you a VERY fine rifle!
Another choice to look at is the Savage bolt action in left hand. With a bit of tuning, they can keep up with anything from Remington and Winchester.
Also keep looking around for the Low/High Walls... I know the problem of the "roll over" cheek piece all too well... but if you keep looking, you'll find one with a cheeckpiece you can live with.
Have you considered a Ruger #1? I know a lot of folks locally have had trouble getting theirs to shoot, but then again, there are a lot of folks HERE that have had no such problems (or at least, they have solved them to their satisfaction..).
My overall plan if I was in the same spot?
Go buy that 40X today... Start shooting it and load developing.
Then pick up a good martini action, and barrel, and slab of wood, and start the process of having a nice custom Martini (or Low Wall or High Wall!) built while you are shooting your 40X.
Just one Lefty's opinion
Paul F.