Slightly off topic, there is an almost 2K difference in the voucher amount between that area and a Congress Heights or Hillcrest. There are some very nice and expensive homes in Hillcrest. If the federal limit has set an amount for a 3 BR I don't understand why a local jurisdiction would overlay their own limits especially with the housing shortages.
Back on topic, a lot of people advertise the small room as a den but as @Russell Brazil mentioned, it is 70 sq ft, a window, closet, and door it is legally a room. The closet is debatable. If you wanted to upgrade, you can switch out the appliances for a stainless steel set that you can get at a Black Friday sale. You can get a decent kitchen package for about 2K. Laundry and parking are also ideal. I think the oak cabinets are OK. Too bad that property manager did not work out but there are other property managers and agents who will do the marketing and present tenants to you. Make sure they know your criteria ex: credit score = X, income = Y times the rent. Source of income is a protected class. If the income is there is there. You cannot and should not discriminate because funds are from a particular source. Remember you are not living there, someone else may be willing to pay the 3800.