1/0 copper - insane (probably great for the electrician's profit margin, though.) 3/0 Aluminum will carry (slightly) more current at FAR less cost, and the terminals on the breakers and panels are aluminum, so it's very compatible with them. Prepare them properly and torque them properly and don't blow huge wads of cash on nothing useful.
I don't find VNTC "tray cable" rated for burial*, so you need a conduit, so it's nuts (the far milder version of insane) buying cable rather than running individual wires in conduit (much, much easier to pull. Also, you can buy 320 feet of wire with a possible bulk discount .vs. 80 feet of cable. But do shop that rather than assuming. Supplier I just looked at had quadplex 40 cents a foot cheaper.)
*I strongly advise against direct burial anyway - the trench has to be deeper, which costs more, the cable is more exposed to damage, and any savings over buying conduit goes away and then some the second time you have to dig the trench (due to the cable you can't simply pull out getting damaged underground, by rocks, rodents, or whatever.) Trenches are expensive, conduit is cheap.
Then we run into the unanswered questions about your load calculation and use case. But those two seem fairly clear, to begin the sanity checking process, while awaiting more details.
An alternative to "further upgrading your newly upgraded service" depending on the nature of the "shop" and its needs for power is to have a second service provided to it directly. If it's for a business, this makes keeping the business and household expenses separate much easier. It usually costs more per month to have 2 meters, but depending on the price of a service upgrade .vs. a new service drop that might be OK (particularly if it is for a business.)