I think the easiest way to make your idea work is to design each room as a separate structural module, but have the rooms share a wall.
For the second room, I would not try to make its centre tree-trunk beam depend on the first room's beam. Instead, both ridge beams could meet at the shared wall, with that wall designed from the beginning to support the beam ends.
That support could be a properly designed timber post, reinforced pier or another structural detail incorporated into the wall. The important part is that the roof load has a clear path down to the foundation rather than relying on cob alone around a beam pocket.
If you know where the next room will eventually go, I would also plan the doorway in that shared wall from the beginning. That way you are not later cutting a large opening into a wall that may be supporting part of the roof.
This could reduce the second room to three new walls instead of four, while still keeping the dividing wall between the rooms. I think keeping those cross walls could also be useful because a very long chain of rooms will need good lateral stability.
The oval shape may make this more difficult. One compromise might be to keep the outside ends and corners rounded, but use straight walls where one room joins the next. You would still get much of the character of a cob building without making each future addition complicated.
For the roofs, I would keep the same ridge height, pitch and orientation for each room. When another room is added, its roof can simply continue the same general roof line rather than creating valleys or complicated intersections.
I would also make the roof overhang on the future-addition end of each room easy to remove. While the room is standing alone it protects the wall, but when the next room is built that section can be altered and the new roofing joined properly.
Your plan for a long southern wall also makes sense for passive solar heating, especially with your strong winter sun. I would still avoid making the overhang unnecessarily deep on that side. A combination of correctly sized overhangs and adjustable window shading may give you more control. This guide on plantation shutters vs roller blinds (
https://iseekblinds.com.au/learn/plantation-shutters-vs-roller-blinds) explains some of the differences in controlling sunlight, privacy and heat, which may give you some ideas even though your building style is quite different.
I would also be careful about choosing the room width only according to the length of the available trees. Timber species, diameter, taper, knots, roof weight and actual span all affect how much a beam can safely carry.
If it were my project, I would spend some money at the beginning getting one repeating structural detail checked locally: foundation, wall, bond beam, ridge-beam support and roof connection.
Once that detail is worked out, you could repeat essentially the same small room each dry season.
That seems to fit your original plan quite well: one manageable room at a time, simple roofs, locally available timber, shared walls between rooms and no complicated roof junctions as the house grows.
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