[Bug] Custom AI models configured in Copilot Chat Agent interface are not selectable or persist incorrectly

Type: Bug

Window: Agents

### Steps to Reproduce
1. Open the VS Code Agent Model settings interface.
2. Add a custom model configuration (e.g., DeepSeek) by providing the necessary API URL and API Key in the custom endpoint settings.
3. Save the configuration.
4. Open a new chat session (New Chat) in the Copilot Agent panel.
5. Attempt to select the newly configured custom model from the model dropdown list.

### Actual Result
The configured custom model does not appear in the dropdown list, or if it appears temporarily, it immediately reverts to the default "Auto" model upon starting a new chat session. The configuration does not seem to persist or be correctly registered for new chat sessions.

### Expected Result
The custom configured model should appear in the model selection dropdown and remain available for use in all chat sessions after it has been saved in the settings.

### Additional Information
- Environment: VS Code 1.131.0, Remote WSL [Ubuntu-24.04]
- The issue persists even after reloading the window.
- It appears that the Agent interface fails to bind or acknowledge the custom configuration globally or within the remote workspace context.

VS Code version: Code 1.131.0 (e4c7e7b1d6d060162f4aa7f8225271b67ce1df75, 2026-07-28T10:51:25Z)
OS version: Windows_NT x64 10.0.26200
Modes:
Remote OS version: Linux x64 5.15.167.4-microsoft-standard-WSL2

System Info

|Item|Value|
|---|---|
|CPUs|12th Gen Intel(R) Core(TM) i5-12450H (12 x 2496)|
|GPU Status|2d_canvas: unavailable_software
GPU0: VENDOR= 0xffff [Google Inc. (Microsoft)], DEVICE=0xffff [ANGLE (Microsoft, Microsoft Basic Render Driver (0x0000008C) Direct3D11 vs_5_0 ps_5_0, D3D11-10.0.26100.8875)], DRIVER_VENDOR=SwANGLE, DRIVER_VERSION=10.0.26100.8875 *ACTIVE*
Machine model name:
Machine model version:
direct_rendering_display_compositor: disabled_off_ok
gpu_compositing: disabled_software
multiple_raster_threads: enabled_on
opengl: disabled_off
rasterization: disabled_software
raw_draw: disabled_off_ok
skia_graphite: disabled_off
trees_in_viz: disabled_off
video_decode: disabled_software
video_encode: disabled_software
webgl: unavailable_software
webgpu: unavailable_software
webnn: unavailable_software|
|Load (avg)|undefined|
|Memory (System)|15.72GB (2.34GB free)|
|Process Argv|--crash-reporter-id 5bc84a5c-35a4-42ff-b02a-6f3124039b1a|
|Screen Reader|no|
|VM|0%|

|Item|Value|
|---|---|
|Remote|WSL: Ubuntu-24.04|
|OS|Linux x64 5.15.167.4-microsoft-standard-WSL2|
|CPUs|12th Gen Intel(R) Core(TM) i5-12450H (12 x 0)|
|Memory (System)|7.62GB (5.28GB free)|
|VM|0%|
Extensions (5)

Name|Identifier|Author|Version
---|---|---|---
x86 and x86_64 Assembly|13xforever.language-x86-64-assembly|13xforever|3.1.5
MySQL Syntax|jakebathman.mysql-syntax|jakebathman|1.3.1
Chinese (Simplified) (简体中文) Language Pack for Visual Studio Code|ms-ceintl.vscode-language-pack-zh-hans|MS-CEINTL|1.131.2026072717
GDB syntax|pierre-payen.gdb-syntax|pierre-payen|0.6.4
vscode-pdf|tomoki1207.pdf|tomoki1207|1.2.2

A/B Experiments

```
vsliv368:30146709
binariesv615:30325510
nativeloc1:31344060
dwcopilot:31170013
dwoutputs:31242946
copilot_t_ci:31333650
e5gg6876:31282496
pythonrdcb7:31342333
6518g693:31463988
aj953862:31281341
82j33506:31327384
envsactivate1:31551504
cloudbuttont:31379625
42190218_ostrepl:31403339
56b7f661:31457055
ec5jj548:31422691
cp_cls_t_966_ss:31526232
4je02754:31466945
c3h7c220:31478652
cp_cls_t_1081:31454832
conptydll_true:31498968
e9c30283:31461165
test_treatment2:31471001
46204921:31447328
capico_a_098a3278:31555650
idci7584:31464702
e3e4d672:31494082
ei9d7968:31496641
chat:31457767
8hig5102:31480529
89g7j272:31518289
7e187181:31503455
i2gc6536:31499202
52612955:31516516
ghj88844:31499326
h08i8180:31475369
ddid_c:31478207
hmra_i5g22:31518061
7df3h592:31512476
cp_cls_t_1082:31535311
logging_enabled_new:31498466
db5d2638:31499441
jb_cp_cls_t_632:31543129
748c7209:31512887
32d76977:31512328
ha629193:31508444
cdk-lw-on:31524445
f9bf7985:31543584
jh5f2457_c:31540921
41836166:31537719
jbcp_cls_pctr_t:31531130
61138546:31518536
cp_intellij_t_nes:31548657
gf14b233:31526830
nes-curr-leftover-1500:31538366
ahp-both-windows:31556933
h561g132:31533256
sessions_5:31532894
ihg5j128:31534457
f412h606:31551416
7g2b5551:31542111
max-result-100:31551154
nes-ex-cf-euporie-03:31542868
5fe6f778:31554331
allow-none:31555437
6i51c363:31551401
c7c27ce7:31554789
nes-gb2-ctrl:31554343
8hc4g144:31558143
0dg73464:31554319
treatment-23-1:31555779
unuse_dynamic_mcp:31555281
session-archive:31558132

```