How many teeth does your clutch and axle sprocket have now?
Yep, you'll need to go with 6" diameter rims for taller tires. The 13 or 15" front tires might end up rubbing on your Bumper & Nerf-bars when making tight turns. Although you could re-position (or possibly add more) wheel-spacers to achieve additional clearance. You could also help this situation by purchasing front rims with the maximum available Offset...
I never tried 15" but my buddy ran 13" tall turf tires on his old racekart, It still turned fairly sharp before rubbing.
You probably have a stack of King Pin spacers in-between the front Spindles and the frames Spindle mounts. You could re-locate all those spacers on top of the spindles for a bit of additional ground clearance.
Your frame probably also has serveral different holes for bolting the rear axle on, that would change the rear ride-height too...
You'll need to double check your Hubs bolt patern and center-hole size to see if you need the Metric or US bolt patern, before buying rims. Here's how to measure: