Why would you want to if your flight leaves at 0710? One hour before (06:10) would be plenty, cut-off is only 30 min before departure. There's not all that much to do in the Budget Terminal after check-in, although you can grab breakfast at tolerable prices (better than on the plane).

As for the rest, Immigration and baggage claim in Singapore usually takes under 15 minutes, although you might want to factor in a little extra as they dig it up from wherever they store unclaimed bags. Taking a taxi isn't worth it unless you have a lot of baggage, as there's a free shuttle bus between T2 and the Budget Terminal, which runs every ten minutes after 4:30 AM according to the official site.

If I was you, I'd snooze happily until 5:30 AM or so, take a quick wake-up shower, and only then mosey over to the BT.