I'm not sure what you consider light but I don't know that I'd say Tikkas are pigs. I think a Tikka with no bottom metal and no trigger is 28oz and 30oz with trigger. If you put a Ti bolt handle on the same Tikka it'll shave another ounce so your at 27 without trigger and 29 with. They don't cost a lot either.
The folks at UM can shave another 5 or so ounces off a Tikka action as well but if you're trying to keep cost down as much as possible you may want to skip that.
To me 29oz w/trigger, 24oz if you have UM lighten it, seems pretty light and for an action that requires no truing can be used as a short or medium action and has great aftermarket support the Tikka seems like the natural choice to build on.
If I recall I've been told the Howa mini, just the bare receiver, is around 10.5oz but I'm not sure what a complete action w/bolt and trigger but no bottom metal weighs. They're quite a bit shorter then a Tikka and pretty much any other true short action so if the cartridge you plan to use will fit in a Howa Mini one of those would probably be the lightest action you can use. I'd make sure and look in to the aftermarket world for the Howa Mini first though because I know they make some pretty light stocks for Tikka's and 700/700 clones and it would suck to have this light compact little action then have to use a stock on it that puts the weight back up around what a LW Tikka build would weigh.