I can't find a purely logical reason why we shouldn't accept a lesser sum, if offered, so long as we agree beforehand not to end the funding search until we meet our initial goal.

Accepting insufficient funding should only happen in a carefully planned way because relationships cost, and every investor will require more work to keep them happy. Furthermore, you don't want to get money and then spend all of it trying to get more money. The money you get needs to be sufficient to move operations forward.

One way of dealing with this is to stage your business plan, and treat the initial "too small" investment as a seed round. Take the small amount of money and get a cut-rate version of your tech to market as fast as possible. See what the market does with it. If it does nothing, fold up your tents and go home. If there's some interest, it gives you more leaverage to pull in the next round of financing. This puts the primary and secondary investors on clearly different footing, which makes managing the relationships with and between them simpler.