You don’t want to use the soldering gun here. Keep it in the drawer. Do you have experience terminating RF connections?

The LMR-600 was way overkill for this application and is going to cause problems for you. I would have stuck with RG-58 or maybe RG-213.

You simply need to terminate both cables with connectors that are compatible. I’d recommend buying a N female connector for Rg-58.

https://www.amazon.com/dp/B09VTBVDMV

Cut the RG-58 short, but not too short. I’d run the LMR-600 to wherever is convenient for it to stop, so that it has zero bends at the top of the mast. Then make the RG-58 long enough to reach the LMR-600 plus a bit for slack. The technical term for the extra slack is: Service Loop. Then buy an N male for the LMR-600.

https://www.amazon.com/dp/B09ZPX1DD8

Terminate both cables and connect them together. You will need to buy a crimper (or dies if you already have a good crimper) to crimp the shields on both sides. You solder the center pins, but do not use a soldering gun. It’s too much heat! Use an iron with adjustable heat.

The big problem is going to be connecting the LMR-600 to the radio. You’ll never find PL-259 terminations for LMR-600. I’d suggest terminating the bottom with another N male. Then take a piece of the cut off Rg-58 and terminate that with another n-female. Then terminate the other side of the rg-58 with a PL-259 that connects to the radio.

https://www.amazon.com/dp/B00K98F2YM

Even if you could find a pl-259 for the LMR-600, I would not use it. Or a straight adapter. The LMR-600 is much too stiff to connect directly to the radio. I’d fear it breaks the SO-239 on the back of the radio. By the time you do all these adapters and pigtails, there is probably almost as much loss as a single piece of RG-213. Plus each termination is a possible point of failure in the future.