Only way: run your own node with non-standard TXs allowed, or pay a miner to include it.
Trust me if I had Bitcoin Core on my computer I would have tried it. I can't help but think there's a way to do it through a push tx service. And I wouldn't want to pay a miner to include it if its malformed, which may be the case. I tried doing it through here:
https://bitaccelerate.com/pushtx/which says this at the bottom:
"Тhіѕ ѕеrvісе іѕ uѕіng thе "ѕеndrаwtrаnѕасtіоn" mеthоd оf thе Віtсоіn Соrе сlіеnt. Yоur trаnѕасtіоn muѕt bе ѕіgnеd аnd muѕt nоt соntаіn unсоnfіrmеd іnрutѕ."
But still got an error.
For reference, this is the unsigned transaction I was trying, would be grateful if someone could analyze it and tell me if something is wrong:
0100000002d89522f8fecd0d83e269909af360551db3810deb2ad7e72a096058dcbe8f2ded0000000000ffffffffd89522f8fecd0d83e269909af360551db3810deb2ad7e72a096058dcbe8f2ded0100000000ffffffff01382d0000000000001976a914fc11decd0e0ab7a332459314069c027328e1dde188ac00000000
This is the error I get from blockstream.info/tx/push when attempting to broadcast the signed transaction:
"sendrawtransaction RPC error -25: bad-txns-inputs-missingorspent"