I have all of the HI-Lift accessories for the HI-Lift jacks.
The problem with the Lift-Mate is that it will lift a tire and rim. You will not be able to take and remove the tire and rim while it is holding up the vehicle. You will still need a jack stand or other device to hold up the axle. On some vehicle you may be able to hook the strap hooks onto the frame or other solid frame component. If the rubber backed jack attachment point was against a solid point you may be able to lift off of that. The rubber backed jack attachment point against the body of the vehicle would crush it in. The rubber backed jack attachment point is designed to press against the tire or tire rim.
The Hi-Lift jack was great in the days of the early utilitarian box shaped vehicles. With the new vehicle with aerodynamics and plastic components they do not work so well.
The OP said that his rear axle is 7,000 pounds. If I remember the spec's correctly the sheer pin is rated for 7,000 pounds on the Hi-Lift. Unless the OP is lifting both tires off of the ground he should not reach the 7,000 pounds.