You can't have two same feeders to an outbuilding
To be more precise, you cannot have two feeders if the voltage is the same. So for instance you are allowed to have a 120/240V feeder, and then a 120V-only feeder. But not dual 120/240 feeders.
Check with your AHJ if you can have a 120/240V feeder and a 240V-only feeder and move all your 240V-only loads there, however there are scant few 240-only loads (water heater, A/C) and 240-only panels are very confusing to "the next guy" who is likely to not realize it's a 240-only panel and land a neutral on the ground bus out of habit.
Option 1, the "right" way:
Abandon the existing feeder, this may be for the best if it is 3-wire feeder. Run a large 120/240V feeder that covers all the house's needs, and make the existing subpanel a sub-sub-panel of what you're installing now. So 2 more spaces needed (see how fast they go?)
Option 2: The "cheap" way:
Downgrade this circuit to 120V-only, and wire the new subpanel as a 120V-only panel. You would run one single hot in conduit, and split it to land it on both lugs of the shutoff switch (which is probably a breaker of some kind). It cannot support 240V loads and must not use MWBCs.
However at 120V, you need to flow twice the amps for the same power - so where a 60A/120-240V service is needed you need a 125A service of 120V only. Or not, depending on your load calcs. Also, at 120V, you have twice the voltage drop, which starts to matter at this distance, so if your expected actual loads will be within 40% of the breaker size, upsize your wire 2 numerical sizes. Also, with wire sizes larger than #6 you should really switch to aluminum feeder, which requires upsizing 2 numerical wire sizes. "You're gonna need a bigger conduit!"
One more thing: if you a sending two feeders over, the old and new panel must be right next to each other. This is due to the "Rule of Six": you must be able to cut all power to the building with less than 6 switch throws, and those must be obvious, and in one location.
The rest of this answer was written before I realized you already had a 120/240V feeder going to the outbuilding.
Reasonable, but... That panel is way, way too small
Far and away the #1 panel question we get is "I am out of panel spaces, what can I do?" The sad thing is, the price differential for larger panels is just crazy small - a couple of bucks per space. Whoever installed their too-small panel, got a few pizzas with the savings. Whoo whoo. And now the homeowner has a $2000 problem.
This isn't a tiny house with trailer wheels under it. At 1200sf it's a real, full sized house, and needs a proper panel. My go-to panel is 42 space, but since you have a mind toward small, consider 30 space, but that is absolutely the smallest I'd go. Seriously.
"I can use thise 2 breakers/1 space double-stuff breakers". Not anymore. Almost every circuit these days needs AFCI, GFCI or both, and those breakers do not come in double-stuff.
Since it is an outbuilding, it needs a main shutoff switch, but does not need a main breaker. The cheapest way to get a shutoff switch is use a panel with a main breaker, as you propose. The main breaker can be of any size, and does not need to be the same size as the supply breaker. In fact, making it the same size is pointless, since it isn't really possible to "sequence breakers" so the local one trips first. Murphy's Law says the main-house breaker will trip first anytime it is raining. :)
So I recommend a 30-42 space panel with a built in main breaker of any size (commonly 200A on a panel with that many spaces)... So the panel has an obvious main, unlike these backfed jobs where you can't tell which is the main. Nice if it comes with accessory ground bars so you don't have to buy those. Also look for packs that include "bonus breakers". Larger panels come with more bonus breakers, that alone can pay for the difference in cost.
That panel you specced is actually a 6 space panel. For your shutoff switch, you specced a retrofittable "main breaker", and that actually is nothing more than a backfed breaker, meaning it sits in the main breaker area, taking 2 of the 8 spaces (And must be strapped down with the THQLRK2 Retainer kit). It would have been desperately small and you would have regretted it the first time you tried to add a circuit. I get where that panel was $20, but not after you add the grounding bars, the "main" breaker and the THQLRK2 kit.
A too-small panel means having to double-up on circuits too much, and making sacrifices like having clump the fridge onto kitchen receptacle circuits, so being unable to dedicate a non-GFCI circuit for the fridge. (Another frequent question we get, "how do I stop my fridge from tripping GFCIs?"). There are so many ways to dramatically improve usability of house wiring and it doesn't cost more than a latte.
And think ahead on that conduit
1" will get the job done today, but will foreclose on ever upgrading the power or using cheaper aluminum wire. It may also be a difficult pull, and where that gets ya is if you find the pull too difficult, you end up having to call the professionals. They won't do just the pull, they'll insist on doing the whole job soup to nuts. Suddenly you have a $1000 electrician bill, but grats, you saved $50 on pipe.
PVC is the cheap stuff, but price it several ways - I would prefer 2" because that is when the price increment starts getting steep. But as a bare minimum, 1-1/4 so that the service could be bumped to 100A in the future.
Do remember comms cable cannot go in the same conduit with mains power, so if you want comms out there, think about dropping a parallel 1/2" conduit into the trench.