Keep in mind that the GP4k is going to be a little oversized compared to the GP5k. In general, newer tires are a little smaller than previous generation tires because they're compensating for the additional width of modern rims. IOW, on the same rims, a 25mm GP4k is probably closer to the size of a 28mm GP5k, so a 30mm GP5k probably isn't going to be that much bigger than your current tires, maybe 1-1.5mm additional on each side (width and height/radius). I would probably get a 6mm allen wrench and use it as a feeler gauge around the tight spots of your current tires. 4mm worth of clearance, to all sides, is what's generally recommended, so if that 6mm spacer doesn't jam anywhere, you're probably good to go.
Oh, and FWIW, some tire manufacturers have started being a little more explicit with their sizing to alleviate guessing games like these. Some will tell you that they'll measure to size on a specific width rim ("28mm wide on a 19mm internal width rim" or something like that) while others have gone all out with WAM (Width as Mounted) and RAM (Radius as Mounted) charts for combinations of various tire sizes and rim widths. I don't think that Conti has jumped on to the WAM/RAM bandwagon, but I they might specify their reference rim width.