Jennifer, if you have the room, you could spread that clay out somewhere, 6-12" deep. It would smother any weeds under it!
Even very minimal amending of the top inch of clay would let many cover crops take root "enough" that their roots would gradually invade and amend the clay, turning it into soil for you over a period of years.
If they were nitrogen fixers, you wouldn't need to fertilize it much. Just give them time.
Clay is a really valuable soil component, holding and exchanging nutrient ions. It only drowns roots when there is too much of it.
Your best clay-tolerant cover crops would be known at local feed stores or extension services, but maybe some of these are suitable for your locale:
Annuals that tolerate and loosen heavy soils, cool weather:
Austrian Winter Pea
Annual Cool Weather Cover Crops
Barley, Awnless
Barley, Bearded
Bell Bean
Clover, Alsike
Clover, Berseem
Clover, Hykon Rose
Clover, Nitro Persian
Clover, Subterranean Mix
Clover, Sweet-Biennial Yellow
Daikon Radiash (??)
Legume / Oat Mixes
some Mustards
Oats, California Red
Oats, Cayuse White
Oats, Montezuma Red
Oregon Annual Ryegrass
Rye, Cereal
Rye, Merced
Tetraploid Annual Ryegrass
Triticale, Trios
Vetch, Common
Vetch, Hairy
Vetch, Lana Woolypod
Vetch, Namoi Woolypod
Vetch, Purple
Zorro Fescue
Perennials that Tolerate and loosen heavy soils:
Alfalfa [P]
Clover, Strawberry [P]
Clover, White [P]
Creeping Red Fescue [P]
Ryegrass, Perennial [P]
Sheep Fescue (Covar) [P]
Timothy Grass [P]
warm weather + heavy soil:
Buckwheat
Cowpea, Red 50
Crotolaria
Lablab
Sesame
Sesbania
Soybean
tolerates poor drainage:
Wheat, Red Winter
Clover, Alsike
Clover, Berseem
Clover, Nitro Persian
Clover, Sweet-Biennial Yellow
Miranda Pea
Rape, Canola
Rye, Cereal
Rye, Merced