Weight; The frame isn't what makes a KLR heavy, it's the 125lb engine. Occasionally people try to pare the bike down to nothing but frankly it's a waste of time 'cause it can never be truly light. That said there is some "low hanging fruit" that can make a difference without wrecking the bike for the purpose it's intended: silencer can save 6 lbs, LiFePo battery can save 9 lbs and there is 5 - 10 lbs in misc "stuff" (passenger pegs and brackets, chain guards, licence plate holder, rear inner fender, etc.) All in all, I've taken off about 20 - 25 lbs off my Gen1's. I also use an IMS 6.6 and JNS rad guard which gives me good rad protection without adding 11lb crash bars.
There are no inverted forks that are a direct fit. Emig racing makes KXF conversion clamps KLR650 - EMIG Racing
but they aren't cheap! You can also have a machine shop press your KLR stem into the donor fork triples.
It's kinda a waste of time and effort IMO; the KLR's weight and chassis prevent it from using or needing a full on MX fork anyhow. Plus there are several issues to be addressed with the conversion; the biggest is a new front wheel/hub and brake system.When I did the math prior to making the decision to stick with the stock forks and go with Cogent's DDC's, my total came to $2,000 - $2,500 for the USD fork conversion vs. about $350.00 and less than an hour for the Cogent stuff.
After the additional expense, fork rebuilding, respringing, revalving, dealing with the rear suspension travel issue, gauge cluster, speedo, turning radius issues......you still have a smaller hub with a smaller axle, smaller bearings and when compared to the SV caliper 320mm rotor mod, a much smaller brake caliper, pad and rotor......all of which are fine for MX use and less fine for dual sport use.
There is no doubt as to the superiority of the modern MX fork.....but unless you have it resprung and revalved properly, it isn't going to work as good as a set of DDC's and springs in the stock forks.......and even if you do those things, the rest of the KLR isn't up to making full use of them anyhow.
This is all just my humble opinion; I'm not against modifying KLR's, in fact, mine are extensively modified for my use. Usually people trying to "make the KLR into something it's not" by swapping motors, suspension, and other major surgery, end up dissappointed and eventually selling their KLR's for a different bike. By the time you do a frame and fork swap (and all the frame surgery to fit a KLR engine in an alum mx frame) you'd be far better ahead to sell the KLR and buy a DRZ, WRR, 500EXC, etc. ....but it's your bike.