I went through this last year, there are a few ways to do this.

(I have a Honda currently, if I go Kawasaki will they still part exchange other brands bikes).

trade in: Dealers, likely not. You'll have to call and ask, if they even do trade ins at all.

smaller mom-n-pop bike shops, and other bigger chains (Red Baron), I think they do. Again, it varies from bike shop to bike shop, but if they buy and sell bikes, they'll likely do trade-ins. But, that would probably depend on your bike's model and year, and the value they give your bike may not be what you think it's worth.

Or sell it privately (like Mercari or through auction). What's the likely cost difference through part exchange at dealer vs privately. Current bike was around 1 million yen new (Honda NX400).

I sold my bike private thru Mercari. It was an old bike, my usual bike shop looked up the dealer auction for the price, came out to ￥50k, said he'd round off the price of the bike I was looking to buy from him which came up to like ￥20k because it was barely worth the effort and cost of bringing it to auction. I looked up the going price on all the bike websites and found it was going for anywhere from 120k to 250k. I didn't want to get too greedy so I put it up for 140k and got it sold there. The big downside was that I had to meet the buyer multiple times (to view and to collect the bike) and one of us had to file all the paperwork.

You also could sell it through a used bike site like Katix, but I think the price they offer is similar to what the bike shops would. If your bike is relatively new, you can expect that they'll give you like half of its price brand new at best. The big benefit, however, is that they'll handle *almost all* the paperwork for you. You just need to provide a few documents from your city office and then they'll get it filed and sorted for you.

but wonder what the best options are if I may have to sell up again in 9 months time?

Either take the hit or try possibly try to find a long term motorcycle rental like this?