Having built several 300+hp Subaru powered Spyder's with 2.5L STi engines I can share some real world experience. First off, you are extremely limited in the space you have to fit a tire without adding flares to the body. IF you keep the rear somewhere around -0.5 to -1.0 camber, you can fit a 205 rear tire. Anything larger will hit both the inner edge of the spring plate (or 4 link bars with DeDion or coil suspension) and will rub the inside of the outer fender lip as well. For this spec of build we've run BFG Radial TAs on the street and Hoosier Speedsters on the track. We'll reference the Hoosiers since they are the stickiest possible at an 050 tread wear. Off they line they stick fairly well as long as you modulate the throttle... where they get really hairy is when you come up on boost... once you come up on full boost, there isn't a 205 tire that will hold. Nothing like running 80MPH in 3rd, shifting into 4th and standing on the throttle while the rear tires just blow off at 100MPH. :-)
Now, we also did an "R" body kit that was wind tunnel tested specifically for the track. This KILLS the look of the Spyder and makes it look much more like a GP/LeMans Prototype style of car. What it does benefit is that you can run 225 (or possibly slightly wider) rear tires. On the last one we built we ran 8s and 9s with 225s. I can say there was little to no negative side effect in cornering. Hard acceleration had slightly improved traction but overall the feel/story was the same. When the car came up on full boost it just roasted the rear tires, regardless of gear or speed.
In short, we have not found a combination that will hold 300hp in the 1400-1500 lbs of a Spyder, but it is still a TON of fun, albeit a bit scary...
As for too much tire, yes it is possible, but I believe these figures will be at the extremes and maximum performance levels. My experience here comes from us running the classic 24hr at Daytona last year. We ran Goodyear Eagle slicks and Avon slicks. We brought a good friend of my father and Chuck's with us, Rick. Rick is a NASCAR engine builder and former NASCAR tire tech. He ran temps and other info on our 904GTS tires at the track. A quick back story, Randy Beck's 904GTS was flared 2" so that he could ditch the factory 10" wide rears and run 12"s. We stretched a 13" slick on the 3 piece 12" rims. What we found was that the wider tires were cooling off too quickly and never getting to optimal temps like the 10" slicks did. Lap times were similar, and the car stuck the corners a tiny bit better in the tighter sections, but slipped a bit more after a longer straight... Doing tire temps when pitting showed a tire that was too cool compared to a comparable 10" slick. Again, this is info from a track that allowed us speeds near 190MPH and driving conditions that aren't easily replicated (and not possible to replicate in a Spyder) but it does demonstrate that "too much" tire is possible under certain conditions. (both brands of tire had similar results)
All that said, you can still have a LOT of fun in a 300hp Spyder, but it is extreme overkill. If you'd like I can put you in touch with a few clients who have this combination. One runs several tracks in CO and has some good insight on having 300hp in a Spyder...