SvtBullet
As others have mentioned, you can try reducing the tire pressures.
I know it sounds weird but RE suggests in the Bullet Owners Manual that riders riding single use a front tire pressure of 18 psi. (Yes, I typed eighteen psi).
They also say the rear tire pressure for riding single should be 28 psi.
(For riding two-up they recommend 20 psi front, 30 psi rear.)
I don't know if the lower tire pressures will keep the tire from trying to follow those terrible rain grooves but it might.
I do know, I just got my front tire replaced with a Dunlop K70 and the guy who mounted it for me inflated it to 35 psi.
The first time I rode the bike after getting the front tire back in place my RE seemed to want to twitch sideways every time I turned a corner. The whole bike felt unstable.
When I dropped my front tire pressure back to 24 psi, everything returned to normal.
While the 24 psi worked for me, remember, I live in Phoenix and we don't have those damn deep rain grooves cut into our streets and freeways.
Since you do have them, give the lower 18 psi pressure a try. It might fix the problem and if it doesn't, a three minute stop to increase the pressure to something a bit higher is cheap.
PS: Back in the 1950's, recommended tire pressures on many of the mid sized British motorcycles like the Royal Enfield Bullets were often down in the 17-19 psi range.
The tubed tires can be quite happy at those low pressures.