Overview
What you have there is a ranch panel. It has two lugs on the bottom made to carry full 200A to a subpanel on the inside of the house.
Wire shortages must be a problem. The wire you have there, #2 copper, is good to 115A. You can only plan to use 115A, but on feeder you're allowed to "round up" to the next available breaker - 125A. (You can't do that on service wire, but, you already got a favorable adjustment, 310.15(B)(7) which allows you to use 4/0AL or 2/0Cu for the 200A run to the house).
Note that the lugs are outdoors and of dis-similar metals, so make sure to use the No-alox goop. That makes it even more important than usual to use a torque wrench to set lug torques. For an inexpensive inch-pound torque wrench that never needs calibrating, look at "old fashioned" 1/4" size beam-type torque wrenches. You don't need a click-y torque wrench in an electrical panel because it's not a cramped space.
115A is * stupendous amount of power in practical terms. As such, I strongly encourage getting a really big subpanel with a lot of spaces, because it can support almost unlimited tools and toys. A 30-space panel wouldn't be excessive. The subpanel must be >= the feed breaker size. A 200A panel is fine. This is no place to cheap out or fall into the fallacy of sunk cost: you'll just be replacing a small panel with a bigger one later, and that's a lot more trouble and money than doing it now before it's all wired in.
Neutral vs ground
As you figured out, you need to separate neutral and ground.
(separately you'll need ground rods at the outbuilding).
You also are required to mark the neutral and ground wires with phase tape to distinguish them from each other, and both ends must be marked the same. This is to be on the cable for life, so do a nice job of tape, or shrink tube is good.
There is no need to mark the two "hot" phases, in fact there are no color codes for hot wires, except "don't use neutral or ground colors").
On the subpanel, neutral and ground must be separated using an accessory ground bar if not already provided. Any neutral-ground bonding screws or straps must be pulled.
At the main panel, neutral goes onto the neutral bar and ground goes onto a ground bar - either the provided one or an accessory ground bar added. Since these two bars are bonded here, grounds are allowed on the neutral bar (not the reverse). However you might run out of space on the neutral bar if you do. If you have run out of lugs large enough, Eaton makes a lug adapter that fits in 2 normal lugs and gives a larger lug. That wouldn't be a surprise on a panel like this.
Hot wires
They go on any breaker up to 125A, and you can buy what's available now and upsize later.
BR buses may have stab limits. These define the sum of the amps on breakers sharing that bus stab. The best practice is to put your 100/125A breakers abreast on the left or right - not across from each other. Unless your circumstances make the chances of simultaneous use virtually zero.
If you haven't made other plans already, save 2 spaces in the main panel for a generator interlock. It's an $80 solution to safely having a generator power anything on the site (not everything at once, obviously).
I don't think a panel like this is the best place for a "whole house surge suppressor". Some people say "put the surge as close as possible to the meter", but those people entirely wrongly believe all surges come from outside your house. Actually your own appliances are a significant threat. And if you're rural enough to have your own dedicated transformer, that blocks surges (being a bandpass filter tuned for 60 Hz). Trapping you inside with your own surges!
