hcaptcha fails with "Invalid site key"

*This issue was originally created by [**@greenship24**](https://github.com/greenship24) at .*

### Background information

- **Dendrite version or git SHA**: Latest dendrite as of 2022-11-12
- **Monolith or Polylith?**:Monolith
- **SQLite3 or Postgres?**:postgres
- **Running in Docker?**:docker
- **`go version`**:built-in to image
- **Client used (if applicable)**: Element, schilidchat, element on the web, syphon, etc.

### Description

- **What** is the problem:
hcaptcha fails to work -
- **Who** is affected:
Anyone using hcaptcha in lieu of recaptcha
- **How** is this bug manifesting:
Response is "ERROR for site owner: Invalid site key" upon trying to register for the server
- **When** did this first appear:

### Steps to reproduce

- Place site key in the recaptcha_public_key
- Place private key in recaptcha_private_key
- Uncomment all fields under # To use hcaptcha.com..... and use the defaults (as in the dendrite.yaml template)

### Additional
I think it *might* have to do with whatever v2 and v3 recaptcha uses which I presume hcaptcha uses v3 now? Just a guess.
I've verified all settings and have done simple tests otherwise where the site key works fine.

Start with the hCaptcha-related settings in the dendrite.yaml template and trace how recaptcha_public_key and recaptcha_private_key are used during registration. Reproduce the registration flow with the documented hCaptcha values and compare the generated widget configuration with hCaptcha's expectations; done means hCaptcha registration no longer reports an invalid site key.

Written by the indexing model from the issue text.