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I'm surprised it didn't come with a pair of spade connectors to attach once you've shortened it. That being said, spades aren't great and I took the time to replace them with bullet connectors when I was moving stuff around at one point. You might not be able to entirely avoid them though (I know I wasn't, I had to a make a few 'adapter' sections). Maybe solder some pigtails onto the rear light and use bullet connectors everywhere else you can?
Yes, you could solder it, but it's a bit permanent and would be an absolute pain. I'd suggest you get a spare connector and use that, at least for one.
Just to check, your front light will have a long wire and a pair of male spades on some pigtails (or maybe on the housing). Your rear should come with an entirely separate cabl (is the one you're showing in the photo?). The long wire from the front light goes to the hub and the separate cable joins the rear to the front.