I assume that you'll be spending time in Kyoto during your 6 days in the area, and you should certainly try Kyoto cuisine when you're there.

As for yakiniku in that price range, you can find it in Kyoto, or in downtown Osaka (e.g. Umeda or Namba), or in the Shin-Osaka station area, or in Kobe. Yakiniku, and other restaurants in that price range are plentiful all over the place. I wouldn't suggest making a trip especially for an inexpensive yakiniku dinner. Just find a place wherever you happen to be. There's definitely no need to make reservations. I would say that there's no need to plan it at all.

Here's a list of yakiniku restaurants around Shin-Osaka station:
https://gurunavi.com/en/reg/pf_osaka...kiniku/rs/srh/

Note that while you can get from Shin-Osaka to Kobe and Kyoto quite quickly by Shinkansen, the areas that you'll arrive in are not the most vibrant in terms of nightlife. The area around Shin-Kobe, in particular, is quiet. You'll need to take another train to get downtown. Kyoto station has a pretty wide selection of restaurants in and around the station building, but it's more "mall" than "downtown" in atmosphere.