Local options are available in the main IDE chat window but permanently disappear from the Agent window dropdown

Type: Bug

Window: Agents

### Steps to Reproduce

1. Open VS Code with local models/agents configured (e.g., via Ollama, AI Toolkit, or custom local endpoints).
2. Open both the main IDE chat window/panel and the specific Agent window.
3. Click on the language model / session type picker dropdown in both windows to compare.
4. Observe that the "Local" options (or custom local models/agents) **are still present and working in the main IDE window**, but they have completely disappeared from the **Agent window** dropdown, leaving only "Copilot" and "Claude".

*Note: Reloading the window (`Developer: Reload Window`) or restarting VS Code does **NOT** restore the "Local" option in the Agent window. Once it disappears from the Agent window, it stays gone permanently until a complete reinstallation of VS Code is performed.*

### Expected Behavior
The model picker dropdown in the Agent window should consistently sync and display the same local models and agent options as the main IDE chat window.

### Actual Behavior
The "Local" option becomes completely missing *only* within the Agent window interface, while still being available in the main IDE window. Restarts and reloads do not resolve this UI discrepancy; only a full reinstallation of the editor fixes it.

VS Code version: Code 1.129.0 (125df4672b8a6a34975303c6b0baa124e560a4f7, 2026-07-15T00:08:44Z)
OS version: Darwin arm64 25.5.0
Modes:

System Info

|Item|Value|
|---|---|
|CPUs|Apple M5 (10 x 2400)|
|GPU Status|2d_canvas: enabled
GPU0: VENDOR= 0x106b [Google Inc. (Apple)], DEVICE=0x0000 [ANGLE (Apple, ANGLE Metal Renderer: Apple M5, Version 26.5.2 (Build 25F84))], DRIVER_VENDOR=Apple, DRIVER_VERSION=26.5.2 *ACTIVE*
Machine model name: Mac
Machine model version: 17.3
direct_rendering_display_compositor: disabled_off_ok
gpu_compositing: enabled
multiple_raster_threads: enabled_on
opengl: enabled_on
rasterization: enabled
raw_draw: disabled_off_ok
skia_graphite: enabled_on
trees_in_viz: disabled_off
video_decode: enabled
video_encode: enabled
webgl: enabled
webgpu: enabled
webnn: disabled_off|
|Load (avg)|14, 13, 7|
|Memory (System)|16.00GB (0.35GB free)|
|Process Argv|--crash-reporter-id 9b9b9c11-9082-4ff5-8d6c-51371795ed1c|
|Screen Reader|no|
|VM|0%|
Extensions (3)

Name|Identifier|Author|Version
---|---|---|---
Chinese (Simplified) (简体中文) Language Pack for Visual Studio Code|ms-ceintl.vscode-language-pack-zh-hans|MS-CEINTL|1.129.2026071318
Language Support for Java(TM) by Red Hat|redhat.java|redhat|1.55.0
Vue (Official)|vue.volar|Vue|3.3.7

A/B Experiments

```
vsliv368cf:30146710
vswsl492cf:30256860
pythonvspyt551cf:31249601
binariesv615:30325510
nativeloc1:31344060
dwcopilot:31170013
dwoutputs:31242946
copilot_t_ci:31333650
e5gg6876:31282496
pythonrdcb7:31342333
6518g693:31463988
aj953862:31281341
82j33506:31327384
6abeh943:31336334
envsactivate1:31551504
cloudbuttont:31379625
3efgi100_wstrepl:31403338
bj468946:31457056
ec5jj548:31422691
cp_cls_t_966_ss:31526232
c3h7c220:31478652
38bie571_auto:31429954
cp_cls_c_1081:31454833
conptydll_true:31498968
e9c30283:31461165
test_treatment2:31471001
46204921:31447328
capico_axe_098a3278:31551526
idci7584:31464702
e3e4d672:31494082
ei9d7968:31496641
nes-extended-on:31455476
chat:31457767
8hig5102:31480529
89g7j272:31518289
7e187181:31503455
i2gc6536:31499202
52612955:31516516
ghj88844:31499326
ddid_t:31478206
hmra_i5g22:31518061
ie3bb177:31543534
7df3h592:31512476
cp_cls_t_1082:31535311
logging_enabled_new:31498466
j0d79568:31499440
jb_cp_cls_t_632:31543129
56dj4588:31512888
32d76977:31512328
ha629193:31508444
cdk-lw-on:31524445
f9bf7985:31543584
jh5f2457_c:31540921
8hfba999:31537718
jbcp_cls_pctr_t:31531130
61138546:31518536
cp_intellij_t_nes:31548657
d7b18187:31526828
nes-curr-leftover-1500:31538366
h561g132:31533256
session_8:31532895
ihg5j128:31534457
f412h606:31551416
7g2b5551:31542111
sandbox-ui-on-2:31543161
nes-ex-cf-euporie-03:31542868
7642ad28:31551796
6i51c363:31551401

```