125amp feed from garage to house, required wire size, PLUS a separate 3-way light switch
While it appears that your AHJ and utility have given their blessing on this setup...

It seems that you are in a situation where your utility has provisioned you with an additional service as they were unable to increase the size of your existing service (as per 230.2(C)(3) and section 4.1 of Xcel's service standards).  You will indeed need permanent signage at all service or building feeder disconnect locations (engraved placards, most likely -- any shop that does engravings for trophies and such can make such for you) that denotes the location of all disconnecting means on the premises that control power to that structure, as per 230.2(E) and 225.37, as exception 2 to the latter does not apply to your situation.

There are some practical downsides to this that you need to be aware of

There are practical issues with having power derived from multiple services in a single building unit (dwelling unit, tenant space, ...), though.  Chief among these is that you effectively must create an electrical "firewall" in the building plan -- only equipment grounding conductors are allowed to cross this firewall, lest you create a colossal current loop/parallel path (300.3(B)/310.10(H) violation) by inadvertently cross-connecting neutrals or hots derived from the two services.

Given the maintenance difficulties this causes (it's very difficult to keep multiple neutral sources from being commingled by accident when cable wiring methods are used, since you can't do something sane like making all the service #1 neutrals white and all the service #2 neutrals striped or grey) , I would not do it this way; instead, I would rearrange things so that the existing 100A service is routed to solely power the garage and the new 200A service solely powers the house.  This limits the concerns about mixed services to the garage/house 3-way circuit (which can simply be run as a switch loop in a separate conduit in this case, with the boxes separated as well), and also allows us to simplify the disconnect provisioning considerably compared to having the house on a split service, as having conductors on the outside of a structure is typically not cause for requiring a disconnect to be placed there.

In other words, we can run the service-entrance conductors (3x 250kcmil XHHW-2 in a 3" PVC, if I were in your shoes, to leave room for the outside chance of a service upgrade, although the 2" PVC will work, and if the existing meter socket on the garage is a meter-main, you'll need a 200A breaker for it and a 6AWG copper EGC in the conduit) from the meter socket on the garage to the house, while the existing service-entrance conductors from the barn are re-routed to feed the panel already in the garage, and the existing 100A panel becomes a subpanel of the new house panel, fed from the new panel in the addition using a 1/1/1/3 aluminum SER cable run and 100A breaker.

Panel selection matters here

For the new panel in the house, there is one cardinal sin that you should avoid, and that's being stingy with panel spaces.  Given that you are feeding part of the house with this panel, a 42 space panel should be adequate, but it is best to leave room for a second panel adjacent to the first in this case.  It will need to be a main breaker panel, of course, and will be configured as service equipment in this case unless the garage has a meter-main, in which case the house panels will both be subpanels.

Getting the details right

There are several details that we must mind when installing all this.  First off, there is a new NEC requirement in 110.14(D) that all terminations must be torqued to specification -- this means you or your electrician will need to use an inch-pound torque wrench or torque screwdriver when tightening down loadcenter, meter base, and circuit breaker lugs.  Second, you will need to use duct seal at all underground conduit termination points as per NEC 300.5(G) -- this keeps any nasty things that manage to get into the conduit from proceeding merrily into the building via the breaker panel.  Third, you'll need a structure disconnecting means for the garage (the existing disconnect may be a meter main, which cannot be reused for this, requiring you to convert the garage panel to main breaker from main lug or install an enclosed circuit breaker ahead of the existing garage panel if it's not convertible to main breaker at all), and to blank out any knockouts in existing equipment that are no longer used as a result of this reconfiguration.

If you're OK with the consequences of two services into the same building...

If you really want to deal with having two services into the same building (i.e. stick with the existing plan you have in mind), and the potential for trouble down the road this will cause, you can still use the 2" PVC, but you'll need to run four wires through it since you are running a feeder, which requires a separate equipment grounding conductor.  The appropriate wire sizing for this 125A feeder is 1/0 AWG aluminum or 2AWG copper, with XHHW-2 insulation either way, for the hots and neutral with a  6AWG bare copper grounding conductor.

As to that 3-way...

I would run the 3-way for the carriage lights in a separate conduit from the main wires going to the house -- if you are on my plan of having the wires from the garage to the house be the sole service for the house, then  you can't commingle service and non-service conductors in a raceway like that due to NEC 230.7, and if you are sticking to your guns, then the NEC 310.15(B)(3)(a) derate would force an upsize of the feeder conductors if any more current-carrying conductors were commingled with them.

As to the conductors and conduit themselves, 12AWG THHNs in 1" conduit is fine (even if you want smart switch shenanigans later, 4 conductors is adequate as one can put the "main" switch at the garage and the "accessory" switch at the house).  This is a good time to use the rainbow of colors you can get THHN in to your advantage -- the ground must be green, of course, but you can say make the travellers blue and the hot/switched conductor pink or the travellers both yellow and the hot/switched conductor purple -- this makes it easier to tell what's what.