Can I get away with putting a slightly wider tire on my AP1 stock 16's and see any improvement in grip?

 

You can but you won't see any increase in grip, probably the opposite, mainly because none of the best street tires are available in 245/45/16. If you are talking about R compound for the track, then yes, you can go up to 225 front and 245 rear Hoosiers.

 

R1Rs and RA1s are available in 225/50 and 245/45- both are good bets for an autocross tire; the R1R can also be used on the street.

I'd say go for it- it adds an appreciable amount of precision and weight to the steering, and quite a lot of front grip (stock are badly undersized for autox IMO).