What you want may not even be possible. Warning possibly long post.
I know how it works with GEA products (FTTC/P).
CP's need to buy GEA Cablelinks (at a cost of over £1,000) to connect the Openreach L2S to their network/backhaul.
They need to buy a GEA Cablelink for each and every L2S they wish to use.
Some L2S are FTTP only.
I've never ever heard of anyone being switched from 1 L2S to another while with a single provider.
WLR3 lines are not associated to individual L2S. They don't connect to them at all. It's the broadband circuit that runs through the L2S.
ADSL is a different ball game and I don't understand the specifics there.
What you call BAAGTH and BAABYV sound much more like OLT names than L2S names (edit: I see BlackSheep has posted confirming they are OLT names).
If it's an FTTC circuit you don't get to choose which OLT you connect to I'm afraid, most definitely not.
If you are connected to the LCLAN exchange then PCP26 does indeed have an older ECI cabinet (DSLAM) which is connected to the BAABYV (ECI) OLT.
Your PCP also has a newer Huawei cabinet which is connected to the BAAGTH (OLT).
The ECI cabinet had filled so they installed the Huawei cabinet.
The chances of you getting on the ECI cabinet are slim.
So it sounds like your provider only has access to a GEA Cablelink that connects back to the ECI OLT, which will be the older 1Gb Cablelinks (The ECI OLT's don't have 10Gb ports).
This could also be full, not just the ECI DSLAM.
It's an odd situation that the backhaul provider (who carries the traffic away from the exchange) only has access to a single 1Gb Cablelink on an older ECI OLT. Unless it's a very small local provider and it's their own Cablelink.
Most smaller providers would use BT Wholesale or Talktalk Business, both of which will have access to both OLT's.
Ask your provider who their backhaul or "carrier" / supplier is.
It won't be easy to resolve what you want, especially if the ECI cabinet is at capacity.
There is no way to check if an individual DSLAM on a cabinet has capacity. All the checkers simply check if your PCP has capacity and it assigns the appropriate DSLAM.
In your case that's the Huawei DSLAM your provider has no access to.