For me it's hard to find anywhere that feels more classically southern than Oxford and the Cotswolds. But you could argue about whether they belong in the SE or the SW. So why not go halves and call it Thames Valley?
Thames Valley I always think of as Reading and points eastwards, but thinking about it that does make sense, as the Thames does then head through Oxford and beyond.
Living in the far south, I've never thought of the Cotswolds as Southern at all; remember the Cotswolds can clearly be seen from the hills on the southern fringes of Birmingham. Partly Midlands, partly West, but definitely not South.
As a resident (albeit not a native, but I'll soon have been here for half my life) of south Oxfordshire (technically the part that used to be Berkshire, but close to the former boundary with historic Oxfordshire), I'd say that Henley is in the south-east and Banbury in the Midlands, but again with no definite boundary where one region is clearly demarcated from the other. I'm not sure that I'd think of anywhere in the county as being in the west, but once you cross into Wiltshire or Gloucestershire, that's west.
Neither a native nor a resident but yes, Oxfordshire is one of those counties which spans different regions. Henley I would place in the Central South and Banbury in the Midlands, just. I find the character of the country changes significantly once you get north of Oxford on the railway: Tackley and Heyford seem like deeply-rural south Midland or even slightly Cotswold in character, in contrast to the hustle and bustle of the line south of Oxford. In the 80s, when XC trains were still in the hands of diesel locos and Mk-IIs, Tackley was called "Tackley Halt", and Heyford still had (possibly replica) GWR (Big Four)-style signage, this section of line really did seem deeply rural and isolated.
So if we didn't have a separate "Oxford and Cotswolds" region I would say that somewhere a little north of Oxford is the boundary between the South and Midlands. Probably a line going through Woodstock and Kidlington.
I still don't know what region I would place somewhere like Stow-on-the-Wold though, if the Cotswolds were not allowed their own region. A Midlands-West fusion, I guess. It would be interesting to see what place residents of the central Cotswolds regard as their "regional centre". I'm guessing Oxford (hence a separate region) but if not, it would presumably be a choice between Birmingham and Bristol, and that would decide whether they are in the Midlands or West.