I think there are many levels to approach this problem. First off the obvious investigate why your org DNS is having issues. This is IT request they should fix that. They should have SLA on this critical service and not fixing it should escalate to management. There may be many reasons why resolver is not working specially in complex multi site setups. This is the best option as it solves this and probably other DNS related issues.

The rogue approach: On other side if you only host service for handful of users that you personally know and you have ability to edit your hosts file, you can bypass DNS completely. This isn’t ideal as it has to be done one every system and in case your IP changes you will have to do it again. It would largely depend on your level of access to system. If you even can change hosts file.

Alternative crazy idea is to host your own DNS. Change DNS setting on your network configuration. Then point your dns to your org dns. Same problem as hosts file you will need to do that for all systems that need connectivity.

Expanding on own DNS approach you could go as far as hosting your own network. WiFi or switch in case you need Ethernet cable connection. You can buy used enterprise equipment for cheap plug it in l, configure to point to your own DNS and anyone connected to your network would have your settings. Of course this is super shadow IT and I would discourage from pursuing that.

Less crazy and rogue option is to use something like tailscale (or similar) which would have DNS (magic dns). You would need agent installed on every client.