The suggestion to mount the converter on the tower near the antenna might have some good results, however, there are two problems. The first is that you will need two coax relays or a transfer switch up on the tower to switch the converter in and out of the circuit. Then you have the potential to transmit into the converter when you are uplinking on uhf if you forget to switch the relay. When you mount the converter in the shack, you can use an inexpensive coax switch.
You could use a single relay with two feedlines. One would be for the converter output which goes to the rx aux ant jack. That line could be lossy since the converter has some gain and the 10 meter output will be attenuated less than at UHF. The other would be your best coax and would connect to the type N port on the TS2000. This would prevent any possibility of transmitting into the converter. In other words, the antenna would be switched to the converter input or directly to the TS2000 Type N connector.
The second problem is that when you have electronics with frequency determining elements (crystal oscillators) mounted outside, temperature variations will cause drift. Having the converter inside stabilizes the temperature and the converter doesn't drift. Mounting electronics outside also requires special attention to keeping things dry. It's just a lot easier to keep things inside and use good quality coax. Heliax is often available at prices not much greater than LMR400 etc. If sealed up, it will last a lifetime.
73, John K8YSE