Not sure where you got your information. I have driven to AK and back three times, including the complete length of the Alaska Highway, and much of it more than once. It is a very good road for its full length. The speed limits are lower than what you may be used to, but the road is a good two lane road. It certainly does not require four wheel drive. Unless you venture off the road, I see no reason why you should need to be worried about flat tyres. (The only flat tyre I have ever had was in the Brooks Range.)
The Alaska Highway is a magnificent trip. If you are going in May, you will be travelling with the huge convoys of motorhomes moving north. A large percentage of these will be travelling with pets. One thing they will all have in common is their - in some cases total - reliance on the information in The Milepost. With more than 12 months to plan this trip, I would start to study it now, and get a realistic picture of the road and services along the way.
The only part which is not an excellent part of the road is the section between Haines Jnct and Tok. Here the road is affected by permafrost, causing frost heave bumps in places. These are marked, and keeping speed down should not worry you, or your pets. The only annoying thing along the Alaska Highway is getting stuck behind a dozen or so slow moving motorhomes, or those who have stopped to view the abundant wildlife.
Patience is definitely an essential. The more time you have the greater the journey can be. I hope to have a chance to do it all again.
Lifey