You probably can hook up both of these batteries together because both are basically lead-acid. (Mixing battery chemistries, like for example lead acid and Lithium Iron Phosphate, is however a Bad Idea.) But I wouldn't in this case, because there's more to it than just chemistry. Lead acid batteries are built differently inside when they're meant for different purposes. A car battery is meant to discharge "hard and fast", while a deep-cycle lead acid battery is meant to discharge "slow and deep". The differences inside that you can't see are both profound and important, and trying to use a car battery as a deep-cycle more than a time or two will damage it severely. Even just once or twice does a small amount of damage. You may also perhaps not know that discharging a lead-acid battery more than about halfway, whether it's deep-cycle or not, also damages it. Lead-acid batteries are in essence obsolete at this point, due to multiple inherent, inescapable disadvantages like this. The only purpose they still serve well is to crank engines, which for reasons it'd take me a long time to explain is a job that lithium still is unsuitable for. My advice is to just go ahead and bite the $350 or so bullet and buy a cheap 100 ah lithium iron phosphate battery from the list of those recommended on Will's website. You can discharge a lithium iron phosphate battery all the way without damaging it, it'll last through probably ten or more times as many discharge cycles, and probably will be good for ten or more years in real-world use. In other words, it'll probably outlive a lot of vehicles and power tools; this is a long-term purchase and should be viewed that way. To get the equivalent amount of power out of lead acid batteries (without damaging them) you have to buy _two_ 100 ah plus a set of pricey cables to connect them, and the last time I checked lead-acid batteries of that size ran about $100 apiece. So, it's really not as much more expensive as you might think to buy lithium iron phosphate. Besides, a lithium battery is roughly half-- sometimes less-- the weight. You're going to have this thing around for years, and will probably find a hundred uses for it (often it's easier to use a solar generator than run a long extension cord for a power tool, for example) besides hurricane prep. So... Again. I _definitely_ recommend lithium iron phosphate, also sometimes called LiFePO4 or LFP.
As for your solar panel... I have never actually used the solar function on my Licitti for reasons explained in the thread above-- I get my free solar input by other means. But... You can do a thing called "overpaneling" to a certain extent if you wish to. A solar controller can only accept so much input power, and that amount is listed in the device's specs. My guess is that if you stumbled across a 150 watt panel, you'd be okay with it-- the built-in controller would simply reject the extra input. But if you go too far in that direction eventually you can overheat and destroy even a top-brand, pricey solar controller. (Don't ask me how I know.) The big thing is that I'd look for a "12 volt" panel of 150 watts or under. "12 volt" here is a nominal term-- it doesn't actually produce 12 volts. Linked below is a moderately-respected name-brand example. Note that "hard" panels like this are much cheaper than the ones that fold up, and much more durable and longer-lasting as well as a general rule. There are cheaper brands of panels on Amazon that are probably just as good that I'd also personally buy without hesitation. But, this is a brand I know and trust. You'll probably want to purchase some suitable "extension wires" with the correct fitting as well, so that you can locate the Licitti in one place (perhaps even indoors) and the panel some feet away.
https://www.amazon.com/RICH-SOLAR-Monocrystalline-Efficiency-Module/dp/B08NDKHP9V/ref=sr_1_3?crid=MPNLQONJY6R0&keywords=150+watt+solar+panel&qid=1687041130&sprefix=150+watt+solar+panel+,aps,107&sr=8-3
One last thing... I have lived in Florida for a few years now, though I'm not a native, and am therefore no stranger to hurricanes. You will probably also want and need a wall-charger for your Licitti, in part because hurricanes block the sun and you'll need a 110-volt charger to recharge from your generator. While any 12-volt battery charger will work for a LiFePO4 battery so long as the battery doesn't go entirely dead, once that happens you'll either need a dedicated, "designed" LiFePO4 charger (I suggest one in the thread above) to restart it, or you can "jump" it from any 12-volt source just like a car. This is a good thing to know and understand going in. I've never had one go completely flat on me, and hear it's pretty uncommon. But it can and does happen. Also note that you may want to purchase an Anderson adapter of some kind for your charger-- again, see the thread above.
Also... I sense a need to keep to a budget here. If you want to go _super_ basic and don't mind connecting a few wires, learning a few relaitvely simple new tricks and maybe a little inelegantly-designed home-made-from-scrapwood ugliness you can save at least a small amount of money by making your own setup out of components. If you skip the solar-- doable, IMO, for emergency and around-the-house use since you have a generator-- you can cut costs. If it were me I'd start off by buying a LiFePO4 battery of whatever size I could afford. (Yes, even under severe budget pressure I'd not use lead acid. The advantages of LiFePO4-- and there are many more than I've listed here, like much-faster charging-- are simply too great to pass up, and lead-acid just isn't a bargain anymore no matter how you measure it.) Then I'd shop for an AC inverter, searching Amazon and browsing You Tube until I learned enough about them to decide intelligently what size I need and know what size the battery can support, and to decide if I really need a pure-sine inverter or not. (Pure sine costs significantly more, but is much safer for appliances and, especially, electronics.) Then I'd buy correctly-sized wires and a fuse to connect the inverter to the battery, use an existing 12-volt charger I probably already have and my generator to keep it topped off, and call it good enough for this year. Then next year I'd add a solar panel or three and an appropriate controller, and maybe (if you're having as much fun with this as I do) another battery and larger inverter as well. The next thing you know, you'll be recharging your tool batteries or laptops or whatever every single day on solar, and you'll be off and running making more and more of your own free power. Even better, you'll be in _control_ of all this. If something breaks, _you_ will be able to effortlessly diagnose and fix it. Not someone else. _You_.
Good luck, and may the swirling death-storms avoid us both!