Everyone is giving wise advice to rent, and if time matters a lot to you it's what you should do.
But if I were you I'd do exactly as you're doing. Now that you've explained you're really just filling in the root balls, plus spreading some top soil, that's well within the capability of an LX. Slow perhaps, but fun every day, and lets you experience your property and slowly work out all the minor slopes and hollows that you can fill, how the sun moves during the day, exactly what orientation you want that house to capture or avoid the sun.
I don't really see where you'd need to fill the tires. Running a FEL with full buckets and without 3ph ballast is really hard on the front axles. An LX is a lighter weight machine than an L, working it hard with the FEL will load the axle a lot. It's not that it won't do it, it is of course rated to do it. But when you run 3ph ballast it acts as a lever - it's pushing down behind the rear axle, so it's lift the front wheels (if you put enough weight on the 3ph the front wheels come off the ground = zero weight on the front axle).
It's not a big deal either way, worst case you take the tires entirely off, dry them out, refill them. I wouldn't do it personally because I don't think you need it, and if I filled them I'd never get around to emptying them again. As I originally suggested, I'd try it without them filled first. If that's fine, then you don't need to fill them. If it gives you problems, you can get a cheap adaptor and fill them with a hose....water is actually fine if you're not in freezing country (and Florida is definitely not that).
Otherwise, I'd agree with your assessment. Lots of people have ideas for how you could do things very differently, but if you're aiming to spread the dirt yourself and to buy a reasonably new machine to do it, the LX sounds like the machine to buy for your needs. It's about the right size, and gives you the option of a mid mower later (which the L wouldn't). If the aim is to keep it forever after, it's about right for that property.