How much power do you need at the cabin?
A park cabin is not an RV, it's a manufactured home and is regulated like a house, not an RV. (unless it was built by FEMA, then you're on NEC 2005 with amendments to delete AFCI requirements).
Appliances include electric stove, wall unit AC/heat pump, electric water heater, lights and outlets. I plan on changing the stove to a gas stove, and I'll be predominantly using a wood stove for heat. I want to add a dishwasher and compact laundry machine/dryer.
So this basically "ticks all the boxes" for an all-electric home, that typically maxes out a 100A service. That is typical of manufactured homes, because it makes the home easier to provision at the site. Compact appliances tend to be electric too, but "compact" does not mean they use any less power. You plan a gas range but that means bringing gas into a manufactured home that was never built for it -- that's a royal pain and a half. I gather you dislike the usability of electric ranges? Look into induction ranges - totally different animal, and you can "try it on for size" with a a $60 electric induction hot-plate which plugs in. (It's limited to 1500W of course, only because it plugs in).
So given that you are already "maxing out" 100A service, I very much think you should seize that bargain and grab that 4/0 triplex + ground wire (assuming your code permits that).
What to do about that panel at the main service?
I gather you take your service "at an electric pole" and you are responsible to cable it from there. And, the installer provided a 200A, 20-space main-breaker panel at the pole. As much of a fan as I am of "lots of spaces" -- that was the wrong panel to use there. It should've been a "ranch panel" (I call them) which have 8 spaces and thru-lugs. But on Square D, that's fine.
Kinda separate from the gauge question is how to wire the main breaker panel for this... they do sell Square D HOML2125 125amp sub feed lugs for about $50... feed a subpanel without protection. Is this the right way to go about it?
No. The 200A main breaker would (fail to) protect the 125-amp-rated sub-feed lugs, so is illegal to use. You would need to use a 125A breaker there (think of a breaker as a sub-feed lug "with benefits"). What's at risk is the bus stabs, which are typically only rated for 125A for both breakers they serve.
Fortunately for you, you are in HOMeline. Square D makes both a 200A (HOML2200) and 225A (HOML2225) sub-feed lugs, as well as 200A breakers. (all of them bulky 4-space affairs, due to the need to spread load across four bus stabs). Any of those are the right tool for your job. Don't sweat the "Internet prices" - they're insane, aka the "I don't wanna sell it" price... because nobody wants to sell electrical gear online.*
What I'm saying is talk to an actual Square D dealer... by which I mean a real electrical supply house. Find one willing to give you good prices and buy everything there so you develop a relationship. Forget Home Depot, which is overpriced on all but the most popular items.**
And there is nothing wrong with installing a 125A breaker now and wait and see whether there is need for the 200A breaker or lug kit (i.e. if the 125A breaker ever trips).
I realize it's farther to trench. But unless you're using expensive Rigid Metal Conduit (6" of cover; 12" under vehicle pathways), you'll need to trench to 20" or 25", and that means you have to rent a trencher and call the Dig-safe number. So the difference is only labor and wire cost. The 4/0 will perform better for you, and won't limit you. In fact the current stand can support an RV concurrently to using your house, which will have spare capacity for an EV charger, which helps with resale value even if you're a "cold dead hands" type.
* Selling electrical gear on the Internet is a nightmare. The vast majority of electrical products are low-value, bulky and heavy - which makes shipping cost excessive for the items' value. And 90% of your customers will be novices, which means LOTS of returns.
** Home Depot is only cheap on the items they think you'll price-check, like 12/2 Romex. (how do they gather intelligence on that? Big Data and their price-match guarantee. They track every time someone stands in line at the customer service desk to ask for the price match. If 100 people price-checked them on 200A sub-feed lug kits, they'd get in line. But no one does lol, so they overcharge.)