Positional Union and Management Negotiators
Under Union negotiations, if the company you are negotiating with states that their first offer is also their final offer, does that end the negotiations? Or do they go back to the table if the Union doesn't accept?
In the U.S. it's illegal for management to not negotiate past its first offer. From what you've described, it sounds like management are being competitive, positional and may not want to negotiate.
If you take their 'position' seriously, then two options include asking them to come back when they are ready to negotiate, or to make a counter proposal.
I would suggest that you prepare a carefully worded 'negotiation frame' to talk about collaborating together for mutual gain, and talk about the mutual pain you would experience should the negotiations reach a deadlock.
You would do well to work fast and hard at preparing alternative plans (BATNA) should negotiations reach a deadlock to cope with the worst case scenario of a strike or similar. This reduces their power and increases yours. So by the same token, figure out what their alternative to negotiating with you is. What will they do after 1 week, 2 weeks, 1 month etc?
Linking and Log Rolling
If management are steadfast and immovable on one issue, then perhaps you can look at gaining ground and concessions from the union on other issues that they are flexible on. This is commonly called 'negotiation log rolling', exchanging or trading across issues/goals.
We would recommend having an exploratory pre-meeting with the union to explore what issues can be linked or added to the agenda. By expanding the list of goals/issues you can often create movement.
Write down your negotiation questions before going back to the table, and ask what is behind their requests or demands. Find out why they can't / won't move, and what they will stand to lose if they do
Negotiators and Egos
Find out whether the 'take it or leave it' offer is supported by all management, or if it's an initiative taken by one or two lead negotiators. If not completely unsupported, then work with those negotiators or members who can help you to loosen their position and open up discussions. Think carefully about the consequences of undermining the authority of their lead negotiators before you go down this path, and orchestrate it carefully. You may choose to give their lead negotiator a bloody nose on purpose in order to swap out lead negotiators if you think this possible and desirable. Positional offers are often made by people rather than as a deal strategy. The people making them often lack negotiation skill and do so out of fear and insecurity.
All too often negotiators fail to build relationships before negotiating on issues. With the right trusting relationship, inflexible 'take it or leave it' negotiation is less likely to occur. Relationships are more often built with people, far more than with teams. So decide who on your team should forge a relationship with the which negotiator(s) from the union's team.
Consider where to hold your negotiation. Two teams sitting opposite each other across a rectangular table is similar to sporting teams about to go to do battle across a sports field. If you can meet in neutral territory, and have smaller teams meet, then you're more likely to make quicker progress towards agreement.