I would vote Nay for the Cassiar that time of year.
In the fall, after some drier weather and some grading, it's not bad with nothing in tow. If towing, avoid it all season long.
The amin route is a little longer, but you join just outside of Watson Lake anyhow ... so most of your fishing will be after you merge. While the Cassiar rout will initally follow the Bulkley, then the Skeena, and then cross the Nass, Bell-Irving, and the Stikine, I'm sure that in June all these streams would be highly blown from spring run-off.
There are some nicer looking streams on the main AlCan before you hit Watson Lake, but I have never once seen anyone fishing them and I don't know what the prospects are ... depending upon your timing, I'd blow through and get back into the Pacific watershed as you near the AK border.
Since you now have a truck like ours, fuel stops won't be an issue with the range of the CTD.
You'll be nicely surpised that diesel is about $1 gallon less than gas throughout most of the way!!
We do it start to finish, 2750 miles in roughly 48-50 hours swapping driving.
The main route has all been paved at one time or another. Construction will show you some of the nastiest roadway you've ever seen though ... tie things down 10 times more than you think you should. The sections are usually short, but memeorable
