Press keys without using the onscreen keyboard

OK, maybe this is a silly problem, but I really want to play the web version of dungeon crawl stone soup on my iPad. I went to the site using Safari and got to the first character selection option, but I couldn't press enter! I don't want to use a real keyboard on my iPad.

• I searched for ways to force the keyboard to appear without a text field, but I got nowhere.
• I tried to find a keyboard add-on for safari, but I didn't find one.
• I even tried to find out whether I could make Siri press keys for me, but they (on Reddit and web results) said no, and Siri said that the functionality isn't set up yet.

I have a few hunches for what might work. I have messed with web pages in the past (a little bit), and I know that you can edit elements of a page from a website and use the altered page to interact with the site.

• I would prefer to simply append a bunch of key buttons onto the bottom of the page so that I can trigger the necessary events. If anyone has some code for that, I'd appreciate it!

• Would it work, or is there a site that has an embedded page at the top, with an on-screen keyboard at the bottom? Could I go to the game on the embedded page, and use the keyboard on the site to control the game?

• Any other (free, digital) workarounds? A different browser? Some app or feature that I missed? Thanks!