Ideas for disposing of tile/concrete waste

Hey, folks! I know this is largely location specific, but we're trying to get creative here and maybe some of you have managed to find a way to navigate your city's system that we can apply to our own. We have recently ripped out a floor in our sun room. We thought it would just be a layer of porcelain tile -- will take a few weeks to get rid of in our normal garbage, but no big. Turns out we had to remove the layer of porcelain tile AND a layer of clay tile set into ~3" of sand mix cement underneath it AND an extra 6"+ layer of cement over yet another layer of porcelain tile underneath that in a small (possibly fireplace-sized) corner of the room. Obviously, this has created an epic amount of very heavy waste. We currently have it bagged up in our garage/back yard, but will be throwing out 2-3 bags per week for several months if we rely on normal garbage collection. Renting a dumpster would require multiple haul-offs due to weight limits, bagster will only take 1 cu. yard of heavy waste (this fits their criteria for heavy waste) and the local dump only takes 1 cu. yard per household and isn't entirely clear on how often they will take that amount. They further will not accept bagged waste and, as some of the only people in Texas without a pickup truck, we kind of need it to be bagged for transport. Our "bulky pickup" happens twice a year, but does not accept "construction waste," which this is per their definitions (and common sense, really).

Any ideas? Had any success in your projects with disposal of large amounts of heavy waste? I'm not opposed to paying for haul-off, I just don't want it to be prohibitively expensive. I saved pretty heavily for this project, but we have limited flexibility in the budget without making sacrifices. I could swing $300-500 without major sacrifices, but beyond that we'd have to leave the floor unfinished for a few months most likely. Everything we've found would likely cost enough to put off finishing the floors for another few months, which puts the whole project on hold. I'll take whatever help I can get -- search terms for service providers I may not have thought of, national companies you've worked with/heard of that provide affordable waste disposal, any insights anyone has into the City of San Antonio, contact info for aliens trying to colonize a planet without access to resources found in busted up floor waste ... anything. Thanks in advance!