Unfortunately, for bad allergies, I would highly recommend consideration of a HEPA filter, and these are not typically reusable with how they work and how ultrafine particles behave. (There are a few washable filter units available, but the ability of these filters to effectively reduce particulate matter over the course of time, especially following multiple washings, is not really well established and efficiency can decrease. If you did want to go this route, Honeywell offers a few products to consider.)

For really bad allergies and cat dander for a medium room, I would highly recommend consideration of a unit with a very high level of airflow that uses a HEPA filter and has a reusable prefilter, as a prefilter that can catch larger particles and be cleaned will prolong the life of the HEPA filter. The Coway AirMega 400s (or smaller 300s) is a fan favorite, because it uses H13 HEPA filters, has reusable prefilters, draws air from more than one side, moves a tremendous amount of air, is quiet, and is energy efficient. Filter costs are not terrible, and you can get about ~15 months from them, especially if you are vigilant with frequently cleaning the prefilter. Medify, Levoit, Rabbit Air, Winix, and Alen also make some sound offerings, and the Alen BreatheSmart, Levoit 600s, and Medify MA-50 and MA-112 are other stars with high levels of airflow, although I would consider the AirMega to have the most favorable balanced attributes in most cases.

If the price of the AirMega is not favorable, the Winix 5500-2, Coway Mighty 1512, and Levoit 400s are ones to possibly consider. They are not as energy efficient and do not have as much airflow as the AirMega, but are still great choices for medium rooms.

Finally, one budget option to mention is the Corsi-Rosenthal Box. This is a DIY filter made from furnace filters of MERV-13 or higher and a box fan. It will reduce particulate matter, although not as well as a true HEPA unit will, and for your case I would definitely recommend a HEPA unit. That said, it's inexpensive, easy, and would likely still help, so it is worth noting.