The Jetbeam Jet II MK is $24 at Fasttech: https://www.fasttech.com/product/5969000

It is very compact but that is because it has a twist switch. A little over budget but you are getting namebrand quality.

Have you considered using batteries that don’t leak in your AAA lights? Eneloops are wonderful.

edit: Sorry missed the twisty part. Good luck getting an e-switch light for under $30 let alone $20. Fasttech has plenty in the $30-$40 range and a few On The Roads in the $20-$30 range.

The e-switch request seems a little strange for a low-use light running primaries. The parasitic drain can leave you in the dark when the light is finally used.