You do realize 3dB requires you to DOUBLE your output power which would be 34 watts into the sending antenna. That's quite a bit more than a couple of watts.there is an issue and we really only need an additional 2-3dB so I do think a few more watts might work.
I like the idea of an extra antenna lower down but connected thru a remotely controllable relay rather than a "T". You could then switch it in on those 1% days. Put it on a push up mast at the site and you may not even need a tower crew if the tropo is bending the path that much lower.
My experiences with tropospheric bending are that the bent path actually delivers a stronger signal than the normal path so a smaller receive antenna could be used on the lower position which might actually make the reception more reliable during the event and not so dependent on knowing the angle of the bending which is probably not very predictable anyway.
Also, if tropo causes another co-channel STL signal to find its way into the main receive antenna, thus causing interference with your own signal, the lower antenna has less of a chance of picking up the interference.
Somebody just has to be near the remote control to make the switch when the trouble starts and when it ends. If you have a dialup remote control you can do that from anywhere. I'm thinking that's not what you have since you say the telephone service at the site is poor.
No guarantee any of that would actually work. Just me talkin'. But if it can be done cheap it might be worth taking a chance on.