So even at peak season, 30 minutes would do...
ご搭乗便出発時刻の30分前までにペットとおでかけサービス受付カウンター（手荷物カウンター内）にお越し ください。
If you are seriously worrying about this, in the pet section, it states that you should be at the "baggage counter" 30 minutes before departure time with your pet.
It should take staff more time to deal with pets than regular baggage, if they say 30 minutes is enough for pets, it is definitely enough for hold baggage.
Especially if you have any status above OWS/JMB Sapphire, you would find 30 minutes at the check-in counter is not the bare minimum, but a waste of time, given that you would have dedicated security lane.
I was already in Sapporo when
zznoname posted this question. So I didn't see this question in time. I decided to book a taxi, only to discover that when I arrived at the airport 85minutes before the flight, the counter wasn't even open yet.
JAL is asking its passengers who are taking JAL3040 CTS-NRT, transferring on to an international flight, to arrive at the counter early. Counters open at "6.30am" (Yes...75 minutes before the schedule take off time).
The international rules of T-2hours (T-3hours at the notorious LHR) simply just doesn't apply in Japan at all.
So sit back and relax, you really don't need to worry until your bus is more than an hour late.