Ground
My question is: how was the original pickups rigged and grounded?
Humbuckers are going to be hotter and probably need a better ground.
Also, I like the idea of taking off the bridge and drilling a hole at an angle FROM under the bridge TO the control at the correct angle.
This might require a long drill bit and very careful eyeballing of the angle do drill at so that the angle is very shallow in order to hit the control cavity and not dive below it and even possibly thru the back of the guitar.
It would be interesting to see a picture of this guitar.
I would suggest being very very careful if you drill a hole from under the bridge to the control cavity. Think it out, use the shortest route, go slow and make sure that your drill bore angle is lined up with where you want it to enter the control cavity. I wouldn't guess on this. I would eyeball it and keep checking that angle as I went, looking at the bore hole from the side of the guitar at EYE LEVEL, straight on so you can exactly see the angle you are drilling towards the cavity at. That is if drilling to the control cavity is even an option on this particular guitar.
Also, long drill bits are hard to find in various diameters if a long one is needed.
I agree that the bridge probably needs a ground wire going to it from the grounding pot where everything else is grounded. This will ensure a good ground. Without it I can see problems unless there is another way it is done on that particular guitar.
The best way would probably be, as suggested, to ground it by laying the ground wire under the bridge plate with the insulation, of course, stripped back and inch and a half or so in order that the bridge plate contacts a significantly long piece of bare grounding wire, pressing it firmly against the wood of the body beneath the plate.
Just looking at my Fender Telecaster HH with the small bridge located away from the bridge pickup with no pickguard, I notice that the control cavity access plate on the back of the guitar is very close to the bridge, just off to the side underneath the guitar.
It would be a simple job to drill a bore from under the bridge to the control cavity - only maybe two inches max. Very easy to not mess up and easily control the bore angle.
If the OP's guitar has a regular tele type control cavity plate cover on top of the guitar in traditional telecaster configuration, then I'm not sure how far the bore would have to be, but you can eyeball it and see if it is going to be super easy to do.
Hope this helps. A noisy guitar is very frustrating and lots of times it is definitely the ground or grounds that are messed up; wires hooked up wrong, wrong color wires going to the wrong place, etc., since there is no industry standardization for pickup wire coloring.