Reddit mentions of Hommii Pet Drinking Water Fountain, 1.6 Liters with Super Quiet Pump and Replaceable Filter, Automatic Electric Water Dispenser for Cats and Dogs, Color Green

Edit: Just watch these videos and hopefully you can learn something


https://www.youtube.com/watch?v=4DyFCiHSEeo

https://www.youtube.com/watch?v=LF9sYIkQSzI

https://www.youtube.com/watch?v=q34H2VhwPjU&t=9s

https://www.youtube.com/watch?v=hjyWXJMrBbA&t=101s

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You could spray him with a spray bottle (they're sold at Dollar Tree (and sometimes Walmart) if you don't have a clean one (with no chemicals/chemical residue))... but it would be hard to make a budgie bathe. I've owned lots of budgies over the years, and only a few willingly do it on their own.

https://www.dollartree.com/colorful-plastic-spray-bottles-10oz/177821

I see you have a water tray already that looks similar to the "budgie bath" that is sold on Amazon... https://www.amazon.com/Classic-Budgie-Bath/dp/B004J1VTK6

You could also consider having something like this:
https://www.amazon.com/gp/product/B01MYWQOE1/ref=oh_aui_search_asin_title?ie=UTF8&psc=1

... But the bird would likely be afraid of it... and since it isn't tame, there probably wouldn't be a good place for it outside the cage anyway.

Birds usually have to feel safe and comfortable to want to bathe