Legs? You need to be more concerned about your arms!
It would seem that those three lenses would be enough.
Sony Rx-100 series. Great pictures, control and easy to use, small.
Big dollar soloution, Sony Rx1. Full frame but fixed lens.
I have a black rapid and regularly carry it attached to the tripod socket on my 70-200 on a D700. (A pretty hefty combination.) I find it balances better connected to the tripod socket. Check your connections and you should be fine.
Often I don't use any strap. Drives my buddies nuts but a) there is no not-concentrating time and b) the strap does not get in my face when I take a picture.
Hold off for a while if you can. Nikon is coming out with three new cameras that may fit your bill.
If you can't wait, I have heard great things about the Sony Rx-100 line.
I shoot Nikon so that may or may not give me credibility...
I bought the Tamron VC etc. for under $900. The Nikon is $2400. So, I got the Tamron, my D700, and the used 17-55 2.8 I bought for my D7100 for the same price as the Nikon lens. Don't get me wrong, I love Nikon lenses but nobody is paying me to shoot.
You will want VC in the gym.
Don't. It's not 1998 anymore.
this is your answer, don't "save as jpeg"