There are not many materials for such temperatures in air. From the range of refractory products there are ceramics of alumina (> 98% Al2O3), magnesia (> 95% MgO), stabilized zirconia (ZrO2 + CaO or MgO or Y2O3) and a couple of other ceramics of pure oxides. The cheapest variant of the refractory bricks, which have quite high strength at 1600°C, is silica brick (dinas refractory, > 93% SiO2). This refractory material is used for furnace arch and walls.
Amongst the materials available in the market, I would recommend you to search for sintered Silicon Carbide. You may find some information on advanced (technical) ceramics suitable for your purpose at: https://www.ceramicindustry.com/ext/resources/pdfs/2013-CCD-Material-Charts.pdf
A potential supplier of SSiC is: https://www.mersen.com/uploads/tx_mersen/22-boostec-sintered-SiC-mersen_09.pdf
I have carried out some oxidation tests of Boostec® SiC at temperatures as high as 1550 ºC for 250 h in air and it showed outstanding performance. It should also perform well under thermal loadings, but I do not have data on its mechanical properties at high temperatures. Hopefully, the manufacturer of Boostec® SiC can provide you such information.
Like Vadim said, Pure Bauxite (Al2O3) would be one of your best options, and if Al2O3 isn't immediately available, you can try pure Kaolin (Al2O3.2SiO2.2H2O). I believe the success behind reaching this temperature can be attached to purified materials like SiO2, AL2O3, Zr2O.