I suspect you could get away with it (ie without the hotel noticing) depending on the size of the hotel, whether or not it is insistent that you leave the key at check-in when you exit for your travels during the day, and the number of days you stay. Nearly all the places we stayed at were pretty small and we were always greeted whenever we came back as we always had to walk by the check-in counter to get to the elevator/stairs. At one hotel, the key was attached to this 6" metal rod which kind of begged you to hand it over the staff as you exit (though it wouldn't have been a problem throwing it in a backpack). Interestingly, at many of the places I noticed that there was a very prominent sign that said no visitors were allowed in the rooms so I'm not sure if that's a clue of how closely they watch who comes and goes.

I don't know how big a budget twin is but the budget single and the budget doubles I've stayed in were pretty tight. I'm not sure how comfortable four in a twin room would be. The wife and I are pretty much "only in the room to sleep" types but we still needed to pitstop for a quick break or freshen up in our room before dinner and we were regularly getting in each other's way unless one was in bed or in the washroom. The biggest slowdown for four in a room however, would likely be rotating through your turn in the washroom (and I suspect it'll be pretty hot and humid by July because I was already getting pretty sticky mid-May). Not only were they small but most of our washrooms had shared water flow between the sink and shower/tub off a kind of T valve at the base of the sink faucet (eg. One couldn't shower while the other brushed their teeth, though I suppose the toilet could be used ).

Although some on this forum seem to have issues with them (do a search), have you considered staying in two doubles at a Toyoko Inn? We ended up paying about 9200 yen / night tax in for a double at their Shinjuku location. Although a bit of a hike to the subway/train stations, they included free breakfast, free in room Internet (with your own laptop) or three free Internet pc's in the lobby, mini-fridge in the room, etc.