Suspension Set-Up for Heavy Cargo

So here's my situation:
I have recently purchased a dirt bike (Yamaha yz250 2-stroke), and my method of transport is with a motorcycle rack on the back of my 05 Forester https://goo.gl/images/bWN7aI <-- That is what I am referring to, a lot of people directly reference towing, when this is related to tongue weight, and the full weight of the bike is supported by the suspension (about 212lbs).
My car can handle this, but just barely. It has my suspension very close to bottoming out in the rear, and lifts the front to the point where steering is "floaty".
Another issue being that the bottom of the rack scrapes when going up driveways, which hasn't proved to be too crucial, but it happens, and in the long term isn't good.
My idea to combat this is to get new springs and struts for the rear and probably king springs lift springs with 1.1in lift , and to either leave the front alone, or get the "oem" king springs which still lift about 1/4 in.
This would obviously have the ride height taller (and possibly stiffer?) in the rear, than in the front, to compensate for the extra weight in the rear.
That is my idea, but I also know next to nothing about suspension, and would like to hear ideas that would be good for my particular situation.
Is it an inherently bad idea to have taller springs in the rear than in the front? Would there be any drawbacks to a suspension setup like this that I should know about?
I appreciate any information
Also, I considered risers, but the issue also concerns suspension travel, which risers will do nothing about.

Check your owner's manual for maximum tow ball weight. With my 2006, this figure is 80 kgs.

This weight relates to both the vehicle and the load limit of the towbar assembly and mounting points.

Exceeding it routinely will not just shag your suspension, but could wreck your towbar and your rear subframe ...

There's nothing wrong with having stiffer rear springs, it's very common. Also having taller rear springs is also very very common, most new cars now have some forward rake. But it's not the ideal solution, you'll still have bad steering & too much weight a long way behind the axle.

Much better off with a small bike trailer

^ good thinking, NL.

That is too much weight on the rear. Motor bike trailer is the only solution
If the weight was inboard it would be a different situation.

The manual seems to say little about tongue load capacity. My hitch is not OEM however, it is a custom 2-inch receiver (only 1 1/4 options are available for this model) with a capacity of 350.
Any information regarding tongue capacity for the vehicle itself has been rather mixed. I've seen some say 300, others say 350.
The rack itself is about 45lbs + the weight of the bike 212 has me at around 260 lbs
I am definitely within the capabilities of the hitch itself, but I can't seem to get a clear answer on how much the vehicle can handle (other than the suspension obviously not liking it too much).
My goal is to avoid a trailer, even a small one, as I barely have room for the bike itself, and a trailer is a whole hassle of it's own.
In terms of how routinely I will be loading my car up like this, we're talking once every week or two if that.
Undeniably a trailer is the ideal solution, but at least for the time being it's not an option.