After receiving a couple of job offers, I proposed a counter-offer to one of the construction employers. Initially, the counter-offer was met with mild agreement, but then a committee meeting was held and the offer was vehemently withdrawn. Did I go overboard?

This is a very tricky area, but to answer your question, the way a construction employer handles counter-offers varies widely. You must judge for yourself if you think they will be receptive to the idea.
It is also important how you approach them with this information. Would you really prefer to work for them or are you just trying to raise the stakes? Some construction employers (read hiring managers) feel threatened and will dismiss the idea immediately.
However, if the construction employer really wants you on board they may meet the other offer. Tell them that you would really like to work for them but that you have to consider your financial future as well (assuming this is true!). The other point to remember is that you don't want to burn any bridges.
There is nothing wrong in letting them know you have other offers you are considering but do so in a way that does not sound like an ultimatum. If they want you badly enough they may raise their offer even before you tell them what the other construction employer has on the table.