After some conversation in my Discord server, we found out that the problem was in the proxy.conf provided by SWAG:

## Version 2023/02/09 - Changelog: https://github.com/linuxserver/docker-swag/commits/master/root/defaults/nginx/proxy.conf.sample

# Timeout if the real server is dead
proxy_next_upstream error timeout invalid_header http_500 http_502 http_503;

# Proxy Connection Settings
proxy_buffers 32 4k;
proxy_connect_timeout 240;
proxy_headers_hash_bucket_size 128;
proxy_headers_hash_max_size 1024;
proxy_http_version 1.1;
proxy_read_timeout 240;
proxy_redirect http:// $scheme://;
proxy_send_timeout 240;

# Proxy Cache and Cookie Settings
proxy_cache_bypass $cookie_session;
#proxy_cookie_path / "/; Secure"; # enable at your own risk, may break certain appsproxy_no_cache $cookie_session;

# Proxy Header Settings
proxy_set_header Connection $connection_upgrade;
proxy_set_header Early-Data $ssl_early_data;
proxy_set_header Host $host;
proxy_set_header Proxy "";
proxy_set_header Upgrade $http_upgrade;
proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
proxy_set_header X-Forwarded-Host $host;
proxy_set_header X-Forwarded-Method $request_method;
proxy_set_header X-Forwarded-Port $server_port;
proxy_set_header X-Forwarded-Proto $scheme;
proxy_set_header X-Forwarded-Server $host;
proxy_set_header X-Forwarded-Ssl on;
proxy_set_header X-Forwarded-Uri $request_uri;
proxy_set_header X-Original-Method $request_method;
proxy_set_header X-Original-URL $scheme://$http_host$request_uri;
proxy_set_header X-Real-IP $remote_addr;

In particular, the directive proxy_redirect. The documentation states: ¹

Sets the text that should be changed in the “Location” and “Refresh” header fields of a proxied server response. Suppose a proxied server returned the header field “Location: http://localhost:8000/two/some/uri/”. The directive

proxy_redirect http://localhost:8000/two/ http://frontend/one/;

will rewrite this string to “Location: http://frontend/one/some/uri/”.

This makes sense, removing this particular line from the config fixes the issue. However, it would be nice to use it by default and turn it off just for 127.0.0.1 locations.

The documentation says: ¹

If several directives can be applied to the header fields of a proxied server response, the first matching directive will be chosen.

So, I made the following change:

 ## Version 2023/02/09 - Changelog:
 https://github.com/linuxserver/docker-swag/commits/master/root/defaults/nginx/proxy.conf.sample

 # Timeout if the real server is dead
 proxy_next_upstream error timeout invalid_header http_500 http_502 http_503;

 # Proxy Connection Settings
 proxy_buffers 32 4k;
 proxy_connect_timeout 240;
 proxy_headers_hash_bucket_size 128;
 proxy_headers_hash_max_size 1024;
 proxy_http_version 1.1;
 proxy_read_timeout 240;
+proxy_redirect http://127.0.0.1 http://127.0.0.1;
 proxy_redirect http:// $scheme://;
 proxy_send_timeout 240;

 # Proxy Cache and Cookie Settings
 proxy_cache_bypass $cookie_session;
 #proxy_cookie_path / "/; Secure"; # enable at your own risk, may break  certain appsproxy_no_cache $cookie_session;

 # Proxy Header Settings
 proxy_set_header Connection $connection_upgrade;
 proxy_set_header Early-Data $ssl_early_data;
 proxy_set_header Host $host;
 proxy_set_header Proxy "";
 proxy_set_header Upgrade $http_upgrade;
 proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
 proxy_set_header X-Forwarded-Host $host;
 proxy_set_header X-Forwarded-Method $request_method;
 proxy_set_header X-Forwarded-Port $server_port;
 proxy_set_header X-Forwarded-Proto $scheme;
 proxy_set_header X-Forwarded-Server $host;
 proxy_set_header X-Forwarded-Ssl on;
 proxy_set_header X-Forwarded-Uri $request_uri;
 proxy_set_header X-Original-Method $request_method;
 proxy_set_header X-Original-URL $scheme://$http_host$request_uri;
 proxy_set_header X-Real-IP $remote_addr;

This fixed the problem!