Here’s my take on this.
I can see the difficulty of the situation, as recovering deleted digital messages can be tricky. On one hand, your most official and reliable option is to request your data archive directly from Twitter. You can find this in your account settings. If the DMs were present when Twitter last archived your data, they might be included in that file. It’s the first place I would look.
On the other hand, if the other user deleted the messages, they are likely gone from Twitter’s live servers for both of you. In that case, recovery becomes much harder. Cache tricks are very unreliable; modern apps and browsers clear their cache frequently, so the chances of finding intact data there are slim. I would start with the Twitter archive and hope for the best.