How wire for main run to a sub panel
A 125A subpanel being fed by a smaller breaker is absolutely fine.  If anything, make sure the panel is big enough in terms of the number of spaces.  More spaces are dirt cheap when buying the panel, running out of spaces is expensive.  I generally go straight for 200A panels even if I mean to feed it off 60A - I'm a bit of a zealot about spaces. 

Your 100A breaker is a wild mismatch for your #8 cable.   Not even close. 

You have several options depending on your circumstances, all radical. 

Change the breaker to 40A

A subpanel is not a service.  Wiring to a subpanel gets its ampacity off the same table as branch circuits, the common NEC 310.15(B)(16).    Since you must pull out of the 60C table, that puts you at 40A.   

Revisit the air conditioner/heater load.  I find it astonishing that a window air conditioner could possibly be 6kw and draw the same as a massive whole-house unit, unless it uses resistive heating instead of heat-pump heating.  Maybe it can work on a 20A circuit.  There is a narrow exception for certain motors, hence ThreePhaseEel's question about your motor nameplate on the A/C unit.  You should shoot him a pic of the nameplate. 

Change cable

If you really want the circuit to be 100A, then upsize your cable to 1/0 aluminum.  (Running large copper such a distance is just a waste of money, and invites dissimilar-metal corrosion since the lugs you connect to will be aluminum.)  

While this is the minimum size for 100A, it will also satisfy voltage-drop concerns as its drop will only be about 2%.  

Beast of both worlds: 100A on the #8

This is beast mode, and not for the amateur who likes living.   Great care must be taken in installation to keep the super-zappy stuff away from loved ones. 

If the cost of replacing the cable is prohibitive, and you need most of 100A, then your only remaining option is to "step it up" for transmission with transformers.  

The power involved is 24,000 watts.  This happens at 240V@100A, 480V@50A, or 600V@40A.  40A is within the current limit of your wire.  

So you get two 25,000 VA transformers that have 600V on the primary (@41 amps) and 120/240 on the secondary (@104 amps).  These things are not cheap, but can sometimes be had used.  Canada's 575V is close enough IMO.   Transformers are bidirectional, so the house's 100A breaker goes to the 240V secondary of one transformer.  This steps up to 600V on the primary, which is attached to the long cable.  The other end of the cable goes to the other transformer's 600V primary.  Its secondary gives 120/240V.

The transformer isolates, so it must be treated as its own separate service, and the neutral must be bonded to a local grounding rod and the grounding system, since the subpanel now counts as a main panel.  Now you can use the service feeder table for sizing your wires (#2 aluminum) between transformer and now-main panel.  

Of the works, only a tiny bit is 600V, and that is isolated, so a person has a chance to survive casual touch.  Still, the installation should be designed to minimize exposure, e.g. Have the wire go straight to the transformer, no splices or reroutes, and put the transformer in an enclosure or outside.