^ Fellowship of the Ring and An Unexpected Journey both had about 7-9 minute film openings that described the backstory of each film's plot and to set things up for the audience. This could've been done for INS. Not much to tell with the Dominion. They did it for the Borg however briefly in FC.
Here is my abridged version of the Dominion conflict monologue
1. Have Picard talking about the principles of the Federation and how they've always been a space exploration society built on peace and unity.
2. As they explored further and further in to the galaxy they discovered a race (changelings) who felt threatened by the Fed. Have Picard say that the Fed always tries to seek a diplomatic solution over conflict, but circumstances at the time didn't allow and first contact was made on the battlefield with casualties for the Fed (DS9 season 2).
3.Touch on how a dialogue was opened with the Dominion (DS9 season 3) but they remained resolute in their ways and the only way they (the Founders) saw forward was the complete subjugation of the Alpha and Beta Quadrants.
4. Rightly ignore the parts about the Klingon-Fed war from DS9 season 4 and 5, ignore the Coup d'état from DS9 Homefront and Paradise Lost but not the Changeling infiltration part, and ignore the Maqui bullshit.
5. With diplomatic talks a failure, open warfare between the two super powers broke out. With the Dominion as conquerors and the Fed as reluctant opponents fighting for truth, justice and the Roddenberry way!
6. Remark that the war as been going on for nearly 2 years and at present the war has no foreseeable end. Set this all to a montage of scenes with music.
The conflict between the Dominion and the Fed really isn't that deep. It could've been done like I briefly outlined. Think of all the Vietnam, WWI, WWII, Civil, and Revolutionary war films there are. Not every single one of them outlines the details that led to open conflict, but they end up being good movies, and audiences like them. Exposition and montage about why the Dominion War is happening wouldn't be lost on the audience if it was done with some style and before the film.
Looking at DS9 for episodes that you would need to gleam info from for a short synopsis of the Dominion conflict would be
The Search 1-2
Homefront and Paradise Lost for the Changeling infiltration parts
The Quickening to show how ruthless the Dominion can be with biological warfare
In Purgatory's Shadow
By Inferno's Light
A Time to Stand
Rocks and Shoals
Favor the Bold
Sacrifice of Angels
The Siege of AR-558 (only if for the desperation of the cause and how it's effecting members of the Starfleet) The air date is close to INS release date so it might not fit.
So yeah for a general audience that has no idea about the who's who and what's what in Star Trek, a brief summation of the important Dominion aspects of these episodes in a 7-10 minute preamble and or montage to the film could have worked.
Like my early post, I suggested we kept the Ba'ku in the film as the indigenous people of the planet and have them caught in the crossfire of the Dominion and Feds war. The audience would see the Dominion as antagonists regardless and the TNG crew wouldn't be able to hold their heads up high over what the conflict had cost the innocent lives of the Ba'ku. The only certainty for the TNG crew would be the war goes on. Leaving behind a trail of lost lives, including the TNG crews on. A metaphor for the life they had on the show and the life they live now as combatants.