Having built my own crane, and having a 5,000# load shift when a strap slipped - I take these projects very seriously. The engineer I was bouncing ideas off of when I built it was giving me a 2K safe limit before finding out I was tying all my columns into the building's walls at 2 points. His concern was always "what if it shifts?" Now that I have 6 years use of that crane, and countless wiggling loads (usually just weldments that aren't well hooked), I've come to realize what he was trying to beat so deeply into my head. We can't plan for best case scenarios. We have to look at what's the worst thing that might happen - will the crane survive it, and will we?
700# isn't a lot of weight in the grand scheme, but it's enough to change your life. I just want what Dave wanted for me: that you stay safe. Crane's are handier than sliced bread. When you get used to having it, you'll use it all the time.
You might be able to sneak some L brackets in on the web of your T and not lose much travel of the dolly. If that joint holds, then the next concern is the short side of the T twisting and spitting the feet fore or aft along the same line as the long leg. Side to side bowing that you're measuring isn't the failure mode I'd be worried about (it might not even be what you're actually measuring either). I'd look at how the short side of the T is reacting to the load and adjust the strategy accordingly. If it's twisting in the horizontal plane, then boxing it or adding another stiffener along it's length would be needed.
I was looking for pics of how my runway trollies are bolted to the web of my bridge to give you an idea about bracing, but I don't have it on the web. I did find the x-ray and pin shots (I'll spare ya the icky feet).
As for your 30' push-pull gun - you need to unfurl it before use. I have a 35' Miller and it's a pain to use because of how much extra leads I have. I tig all my aluminum instead. I've probably run a pound of wire through the PP rig in 5 years of owning it. When I move and start doing dock repairs, I expect it'll be a lot more useful.