You cannot broadcast an incomplete transaction and have another user modify it later, because an incomplete transaction is not valid and invalid transactions do not propagate on the network.
You could share an incomplete transaction out-of-band, e.g. on a message board, via a Partially signed Bitcoin transaction (PSBT) iff the inputs and outputs of the transaction creator are sufficient to enforce their expected outcome by using a SIGHASH_SINGLE | SIGHASH_ANYONECANPAY construction. The second participant would then be able to fill in the remaining inputs and outputs to achieve their intended outcome in addition.
However, if you want two participants to both have input on one output script, your protocol will require at least one round trip, as the original transaction creator cannot sign off on an incomplete output script, and therefore either the second participant would have the ability to unilaterally determine that output or the transaction would have to go back to the creator after the second participant adds their modification.
I must admit that I don’t fully understand what you are trying to achieve, and therefore may be missing a workaround or trick, but the above should cover the general situation. Your problem reminds me of how inscription market places use PSBTs to offer NFTs for sale, and if you are not familiar, it might be interesting to study that approach.
If you generally want to better understand how HTLCs work, you may want to take a look at Elle Mouton’s excellent HTLC Deep Dive, her other blog posts may also be interesting in the context of your line of inquiry.
