Good luck finding something. I have had the same problem. I did finally find a 49.00 Bissell at Walmart, and it actually will vacuum that rug. I bought it for some outside carpet areas, where I dont bring my big vacuum cleaner, and thought, to try it on this thick pile one first, and was very surprised when it actually worked.

I have this vacuum and I can easily push it on the shag rug in my bedroom. The ultra plush feature makes it easy to push on high pile carpets.

https://www.homedepot.com/p/KENMORE-Pet-Friendly-Pop-N-Go-UltraPlush-Bagged-Canister-Vacuum-Cleaner-BC4027/308593452

I believe the Carpet and Rug Institute has some guidance on models that work with plush/soft carpet. We have an Oreck Magnesium RS upright that we bought at Wal-Mart for under $200 to clean a frieze rug. It works pretty well but I don't think it's the best machine out there by any means. Our Miele C3 Marin and central vac also do a great job but are a lot more $$$.