Adapter vercel split option breaks form action

Not sure if it might be related or not but I encounter another issue that happens in the exact same conditions.
Sadly I've been unable to find the time to create and deploy on Vercel a reproduction. Sorry if this is useless noise 🙏

Issue:

In a form action, I trigger an internal fetch call to a +server.ts route handler.
This route handler returns json data (return json({ ... })).
The route handler crashes with TypeError: immutable (full error below), and the action receives { message: 'Internal Error' }. Despite what the error message implies, I'm not trying to either set cookies or headers anywhere.

Conditions:

• Only happens when deployed on Vercel
• Only happens if the route handler is deployed separately (marked with split: true)

Error details

TypeError: immutable
    at _Headers.append (node:internal/deps/undici/undici:2105:17)
    at add_cookies_to_headers (file:///var/task/vercel/path0/apps/client/.svelte-kit/output/server/index.js:2320:13)
    at file:///var/task/vercel/path0/apps/client/.svelte-kit/output/server/index.js:2649:9
    at process.processTicksAndRejections (node:internal/process/task_queues:95:5)
    at async helpersHandle (file:///var/task/vercel/path0/apps/client/.svelte-kit/output/server/chunks/hooks.server.js:1181:18)
    at async respond (file:///var/task/vercel/path0/apps/client/.svelte-kit/output/server/index.js:2638:22)
    at async fetch (file:///var/task/vercel/path0/apps/client/.svelte-kit/output/server/index.js:2388:26)

Environment

    "@sveltejs/adapter-vercel": "3.1.0",
    "@sveltejs/kit": "1.30.4",
    "svelte": "4.2.12",

And Node.js 20.x on Vercel.

Originally posted by @benjaminwaterlot in https://github.com/sveltejs/kit/issues/11879#issuecomment-1971480386

Start with the stack trace in the generated .svelte-kit/output/server/index.js and hooks.server.js, then inspect the adapter-vercel path used when the route handler is deployed with split: true. Reproduce the Vercel deployment with the listed SvelteKit, adapter-vercel, Svelte, and Node.js versions; done means the form action's internal fetch receives the route handler's JSON response without the TypeError: immutable failure.