Your first problem is that 28 September is a Sunday and there are multiple places where buses replace trains.
The nearest you can get to Kemble is Cheltenham Spa from the north or Swindon from the south. After that it is a bus to Gloucester, followed by a bus to Kemble. If you leave Harrogate at 9.15, You get to Cheltenham at 12.52, but then it takes another 3 hours to get to Kemble. By road it is 26 miles, and takes 40 minutes to drive. But the replacement bus takes you to Gloucester, from which another bus takes you to Kemble.
It rather depends on your arrangements at Kemble. Are you visiting friends who might be willing to collect you from Cheltenham?
Your second problem is that on 3 October CrossCountry trains may be on strike, which would mean travelling via London instead of Birmingham.
There's nothing wrong with planning to travel that way anyway. There is a direct train to Harrogate from London King's Cross at 11.33, which you catch by leaving Oxford at 9.32. Arrive in Harrogate at 14.24.
Let's see what the best deals are on basis of Harrogate to Cheltenham Spa on 28th, and Oxford to Harrogate via London on 3rd.
Outbound the trip I mentioned (3h37m), the forum's site offers split advance fares for £132.70, but if you didn't mind sitting at Wakefield Westgate for 2 hours and getting to Cheltenham 2 hours later, it could be done for £81.01. Difference could pay for the taxi from Cheltenham to Kemble and still be quicker that using the replacement buses from the earlier train!
For the return trip there is also a trade-off between journey time/number of changes/fares.
For the journey I mentioned before (Oxford to Harrogate, 4h52m duration), the forum's web site offers split advance fares of £101.85 for 2 railcard holders.
Alternatively you leave 2 minutes later at 9.34, change at Didcot and Leeds as well as in London, have a journey time of 5h59m and arrive at 15.33, but the fare would be £77.37.
That's my best effort. Best of luck!
