Device is reset when switching power sources
When you turn on the PC, the voltage jumps from 3V (or less) to 3.3V. That jump may be detected by the reset circuit in the weather station. Connect a resistor between the PC and its diode and connect a capacitor in parallel with the weather station. When you turn on the PC now, the capacitor will slowly charge to the higher voltage and hopefully the reset circuit won't detect it.

The capacitor can be big (1000uF or more for example), the resistor has to be big, but not big enough so as not to drop too much voltage. As the device uses 1mA, a 100Ohm resistor will drop about 0.1V, which should be OK.

As for your other problem - you should use a relay that is powered by the PC. When you turn on the PC, the relay energizes and switches to the PC. When the pc is turned off, the relay will turn off too and will switch back to the battery. The capacitor will hold enough power for the transition.

There should be relays that can be powered by 3.3V. If not, get a 5V relay and connect the coil (just the coil, connect R1 to 3.3V as usual) to the 5V rail in the PC.

A schematic including both of my suggestions. In this, the R1 can probably be up to 500Ohm as there is no more voltage drop from the diode.