Bear in mind that many Tay beats are multi method, with Harling used a lot on the lower beats, fine if you like that sort of thing - but personally I don't. It makes it a bit more difficult to isolate the fly caught fish to do a like for like comparison with other fly only rivers.
After 15 September the Tweed reverts to being fly only for the rest of the season, which is the main Autumn run, this being the prime time on Tweed.
The beginning of October would normally be best on the middle river between the confluences with the Ettrick & Teviot rivers [Main tributaries & salmon rivers in their own right] or on the Lower river this being below the Teviot down to the confluence with the river Till. Below this the bottom river beats may also be productive in low water conditions, but good running heights will often see the fish speeding through at this time of year. Note almost all of these beats are expensive but get the right conditions & they can be very productive indeed.
The upper river above the Ettrick will need water to fish at this time of year & it is a bit early for this part of the river which usually tends to reach its peak at the end of October & through November, however the beats up here are much less expensive so if you are wiped out by flood or drought then it's a bit less painfull.
Have you considered trying to take a couple of days on 3 different beats on different parts of the river? this may increase your chances of hitting appropriate conditions on at least one of them & give you the opportunity to vary things a bit & see more of the river & the varied nature of the beats.
This link will show you what beats are available on FishTweed:
It also shows the cost per rod/ day, VAT [Tax] is 20% for those beats which charge it, but check up on this as being an overseas visitor you may be exempt [I don't know if this still applies].
Another possibility would be the Lower Dee as the season has recently been extended to mid October & some very good catches are made on the better beats at this time of year. A lot will depend on your budget [as ever] but feel free to send me a pm if you would like to discuss it in more detail [I'm on the Tweed a bit later for a week at the start of November & have been every year for the last 20 odd years so I can give you my impressions but I won't pretend to be an authority on it, just somone trying to get good value for the outlay].
PS - I just opened the link to check it & it gave January rods [close season] so go to the find fishing link on the left, click on it & then click the month you want to open up the availability. I don't know why it did that? T.