Do you have access to the clean out cap? Have you tried running a snake down the drain or in the clean out. A lot of times there will be gunk, especially down the washer drain because of all the fibers, lent and different things coming from the washing machine. I have heard of people using drain bladders but claim that they only work for so long and a lot of the times they will burst. Although, depending on your problem, your drain might keep clogging overtime, even if you clean it out really well. One thing to help would be installing a mesh liner (after the clean out) that catches the link coming from the hose. You will be surprised at how much link will actually build up, even in a week to a month. Also, another method could be installing a laundry sink, this method is kind of similar to what you are referring to about adding a 10 to 15 gallon tank (which might be fine by the way) depending on how your landlord feels about additional plumbing or add ons. Hope this helps