But is it safe to do this? Replace fuse with one that has shorter blow time but higher current rating?
Often you can do this, but you still have to size the fuse correctly. Overrating it if using a fast-blow in place of a normal/slow-blow or a normal in place of a slow-blow is correct, but going up from 0.5A to 6.3A is far too big a jump.1
Since you need to buy fuses anyway, I'd stick with the rated fuse. If you go for a faster-blowing type, then perhaps 600-750mA.