35 amps is the max current rating of the rectifier not the output current. I'm not sure what the rating of the stock rectifier is but I would venture to guess the only thing you'll accomplish by doing this is risking damaging your board with unnecessary soldering.

Actually looking at the stock part it's a DB3501 which is a 35A 100V rated part. So you'll literally gain nothing.

Its not the rectifier that's the issue, the problem you are seeing is the transformer is not able to support the additional load.

I have no idea what you have in mind or what the schematics look like but if youre adding a load, could an arcade style power supply work as a secondary power supply. Run the mods only off that?

I'll try to get you more information later, I have to head out now to get my practice in for the state championships next weekend.

Could you be more detailed on what you're actually trying to do? That will help find a solution.

Challenge accepted.
I'm sorry but "higher rated" - how is this higher rating achieved?

Higher voltage?
If part was rated too low then this can be an improvement. Going to a higher voltage gains nothing if the bridge was sized properly to begin with. I have yet to see one that actually had too low of a voltage specified unless the bridge was replaced with a lower voltage part during repairs. Nobody used anything lower than 100V that I have ever seen. 100V works well for most but pushing threshold for coils. 200V usually used there. I tend to only go 400V since the cost is the same as a 50, 100 or 200V part. Does it gain anything? No, but if replacing the bridge and the cost is the same then why not.

Higher Current?
Higher current rated bridges can handle current surges better but in the long run - you usually have the same power loss and heat dissipation. And in the case of the cheaper bridges (including full metal body bridges) - it is often worse due to cheaper diodes with higher forward voltage drop resulting in higher power loss. And for most machines - you are already using a 35A bridge.
I sold the top of the line Diodes Inc Bridges for a long time (until they quit making them). Were they an improvement over the originals with same current rating? no, not really. They are merely just a replacement for what is there.
Going to full metal body bridges tends to be an issue due to one fact: Look at the surface of that bridge, especially where it attaches to a heatsink. Got that nice and very smooth surface required for attaching a heatsink? No, instead they have that rough, cast look and texture to them. This impedes thermal transfer between bridge and heatsink. The heat slug type bridges (phenolic body) tend to have a MUCH better heatsink mounting surface. The Diodes Inc GBPC3504W bridges were pretty much mirror smooth. Today's On-Semiconductor/Fairchild bridges aren't nearly as smooth but still not bad.

Unless you are drawing near the rated current of the existing part (>50%) and the existing part is rated at 25A then going to these 35A, 100V bridges won't help. The biggest improvement to a bridge rectifier circuit is drawing heat away from the part. Buy dedicated heat sinks for bridge rectifiers rather than using the low power transistor heatsinks. A popular heatsink used by many people for bridge rectifiers is the Aavid 563002 folded aluminum type heatsink which is actually intended for transistors. These have a thermal resistance of 13C/W. If a bridge is dissipating a light load of 5W (often higher) then the heatsink related temp is rising by 65C. Add that to 25C room temp for 90C surface temp. Smallest dedicated bridge rectifier heatsink (folded aluminum) is rated at 9.4C/W. Same load would result in 47C temperature rise. Add 25C room temp for 72C surface temp. Nearly 20C drop just by changing to a similar but proper heatsink. Imagine how much cooler it can run with one of those massive extruded heatsinks.

* It's usually not the bridge rectifier, it's how you get heat away from the bridge rectifier *

And don't even begin to mention the so-called 50 amp versions of these bridges. It's nothing more than specmanship and BS. Under the lid - it's pretty much the same part.

Would it be possible to use something like an Arduino or Raspberry Pi to sense the voltages on the GI and control the LED strips with a separate power source?

https://www.mouser.com/ProductDetail/653-G2R-1-AC6?r=653-G2R-1-AC6

Why couldn't you use one of these tapped off the GI circuit?

Have you tried pinstaduum lights?

It was a joke. Those things are awful.

Thanks for posting the fix, I'm shocked a 35 AMP bridge wasn't cutting it. Yo'll have to post a video of the lightshow!