Hey Dart you've pretty much got it.
Before doing your actual repotting make up a couple different sample size soil mixes put them each in the tiny pots you plan to use for the Lithops. Water the empty pots of soil and set them where the Lithops will be growing (under the same light, temps, etc) and wait a few days then use the one which is totally dry or driest you can get throughout (not just the top inch or so being dry) in about 3 days.
It'll only take a few days to find which mix will be best for you in your area. The higher your relative humidity the less water retaining your soil needs to be mine is 25% RH now and will be up to 50% in August while my main Lithops man's is 6-10% RH almost year round! Hence why he can use all the peat he wants to help retain moisture. So run a few experiments before doing the repotting to find the mix that dries out fastest for you. These plants can be repotted anytime of the year, no buzzers are going off, there's no rush.
In my experience peat + sand is almost like a recipe to make mud (stays wet, gets compact & has no air flow through the soil) that's what folks grow Sundews, Utrics and such in right? That's a swampy soil that doesn't dry out for several weeks in my area (and actually kills my CPs too so I don't use it for them either). But maybe you have larger grit sand than is available here. 1-2 mm grit sand is a good size for "sand" the play sand is bad news my my exp. Way too fine and that's all I can get.
After you have your soil mix pop the peat cube out of the Lowes pot and crush the dry peat cube with your hands, it'll crumble pretty easy, using your fingers tease all the peat out of the roots, there's not usually very many roots on a lithops so it's not very hard to do. When you get all the peat out fill your pot 1/2 way with the new soil and tilt at a 45* angle so you have a soil slope, lay your lithops on the slop and shovel in the other half. Then hold the pot upright and adjust the lithops to upright and tap the pot a few times on a table so the new soil compacts in around the roots. I bury about 1/2 of the body but if you've used a lot of peat in the new soil mix don't bury the body in anything but top dressing (i.e. pea gravel/aquarium gravel, non compacting sand, etc) if the bodies stay wet from contact with the peat they'll discolor and rot.
I don't usually water for 2-3 weeks after repotting succulents as many times plant that were repotted then immediately watered have rotted on me. I lost a new Pachypodium bevicaule that way recently because I thought it looked a bit too desicated after it's trip but my "kindness" worked against me. Waiting a few weeks gives the roots time to heal and recover their strength and the plant a chance to reanchor itself. Don't flood the pot on your first watering, just do a light watering, water doesn't have to come out of the bottom of the pot, if you saw some go in the top that's good enough for the first time, a week later do another deeper watering (water can come out the bottom) then leave them be until they start retreating into the soil and getting very wrinkly, Lithops will wrinkle slightly on warm days but on cooler days they pop right back into shape, but if they don't puff back up and just get more wrinkly then it's water time. It's pretty hard to starve these things for water.
You will want less light hours and cooler temps for winter so your plants know what time of year it is: when to flower, when to start pushing up a new body, etc. They might make new bodies even without photoperiod shifts, a perpetual vegetative state but I've heard they must have a temp and photoperiod shift to initiate blooms. Since these come from a rather temperate climate (S. africa, Namibia, etc) don't subject them to blazing hot temps and direct sun all day, if they turn white you've boiled all the Lith out of your Ops!