How do I best integrate my sub into the LX 300.1 setup?

The biggest thing to nail down first is the ground connection for the entire system. If the ground loop is long or has high resistance, you'll get hum and muddy sound regardless of what mode you choose. Get the negative terminal wires as short as possable, run them directly from the sub terminal to the amp ground point, and make sure that ground wire is massive. It's not just about the power wires to the subs; the return path is just as important for clean signal. As for the amplifier mode, dual mono is almost certainly your best bet for maximizing clarity and controlling the output across the frequency spectrum. Bridging four subs through a single channel amp setup is going to put massive, unregulated stress on the internal components and can lead to poor sound quality and eventual failure. Dual mono splits the load and allows the amp to manage the power more precisely across the drivers. Since the Solo X 8s are designed for specific performance, keeping that signal clean is key to getting the punch you want without sacrificing definition. Wiring the four subs to the LX 300.1 needs to be done according to the amplifier's manual, but genrally, you want to match the impedance load the amp is rated for. Don't try to run them all in parallel unless the amp is rated to handle that specific low Ohms load with sufficient wattage. It sounds like you're using a dedicated sub amp, so check the manual for the exact wiring diagram for driving multiple drivers. The goal is always stable operation at the amplifier's rated power output. Make sure you're using quality, appropriately gauged wire for all the connections. If the wires are thin, they'll introduce impedance issues and limit how effectively the amplifier can deliver that deep, clear bass you're after.

Before you hit that power switch, put a multimeter across the speaker outputs and see what load you're actually presenting to the amp. I learned this the hard way when I wired two subs in a box I built years back. The label on the terminal cup said one thing, the actual coils said another, and I came close to cooking a perfectly good amp on day one. Took me two minutes with a cheap meter from the junk drawer to catch it. Also check what impedance the voice coils are wired at before you decide series or parallel, the coil config on those subs gives you options and the wrong combo is what makes bass sound muddy.

One thing to sort before you power up is the subsonic filter on that amp, since a ported box unloads hard below tuning. I run a ported enclosure myself and set the filter a few Hz below my box tuning, keeps the cones from flopping around on low notes. Also check your port velocity while you're at it, if it chuffs audibly at high volume the port is too small and that's where alot of that muddy sound comes from. What frequency did you tune the box to? That changes a lot about how punchy the low end actually feels in the car.

Update: the multimeter tip saved me. Measured across the outputs before hitting power and the load wasn't what I expected at all, so glad I checked. Still stuck on the subsonic filter though, I honestly don't know what my box is tuned to since I built it from a random plan online. How do I figure out the tuning frequency so I know where to set it?