I researched into this exact question not too long ago.
Generally 125 gallon is regarded as the biggest tank you can keep safely without added floor support. Of course there are MANY factors that can come into play though.
But to keep things short, in your case a 125 gallon should be comfortably safe. Clearly you've done some homework on this so you do have things going for ya, which you mentioned (1st floor, against exterior wall, perpendicular to floor joists). I would say you could even go up to a 150 gallon with no problem, maybe even a 180. But again there are many factors though, so no one can say for sure.
A major sign of weight overload will be the sound the wood/house makes when it's about to snap (like a tree going falling), then you for sure have a major problem and need to drain the tank instantly.
But a sign to check for to see if the tank weight is slowly causing problems is to look if the floor sags (wall moulding will sit higher off the floor since floor is "sinking") as well as things Diana mentioned.
But it is not too hard to reinforce the existing flooring rather than buy expensive floor jacks if needed.
I had a 90 gallon (4 ft footprint) in the center of the house (house has cathedral ceilings so not sure how much different floor framing is) and after research, upgraded to a 150 gallon (only a 5ft footprint, not even a 6ft, so I could actually go to a 180 since the 6ft would distribute the more weight over more area). It does have a flat base stand though, not just a 4 legged stand, although shouldn't really make too much of a difference.
And that's not even the only tank in the room, got 6 others (55, x4 20s, 10)