Can I use a regular gym, yoga, or mattress pad instead or a pole crash mat?

No this is not recommended.  Most gym mats are designed more gymnastics moving across the floor while pole dancing is distinctly different in that pole moves are done up high off the floor and a fall typically happens straight down with a harder impact.

More cushion and protection is needed during pole dancing activities for best protection.  Gym and yoga mats are not designed for that.

Here is a great article that walks you through choosing a great crash mat for pole fitness dancing: https://polefitnessdancing.com/best-pole-dance-crash-mats-for-home-use/