If I place a 4x4 across between the landing gear supports with the jack in the center is the structure of the frame strong enough to support it?
The trailers frame is strong enough but I doubt a small 4x4 6'-7'long will be heavy enough to lift 3000 lbs by jacking in the center of it. It will probably break pretty quick as the frame is maybe 5'- 6' apart. I would use two jacks placed under the timber close under the frame rails instead of just one in the center.
If I understand you correctly the 4x4 will be up against the trailers 2 main frame rails which won't hurt a thing. Don't use a cross member as they're just thin rolled steel.
This was common year ago when campgrounds were all basically unimproved. We dug holes in the ground on the high side and blocked the low side to get a camper level.
We've camped with the landing jacks all the way down or and all the way up just to get level.