Both the Freedom Caucus and the Dems will try to barter votes for concessions. The GOP will be able to play them off against each other and lower the degree of concessions.
This dynamic was first apparent when the GOP were late again getting to the appropriations bills last September. Dems helped them out by providing votes to pass a clean continuing resolution to keep funding as is. In December, it was the Freedom Caucus' turn to provide the votes, with a clean bill again being passed.
Neither the Dems nor the Freedom Caucus gain much in these clean bills. Many of them want to use their leverage (their votes) to gain concessions. Dems want protection for DACA young people (children brought to the US as illegal aliens). The Freedom Caucus has a big roster of wants, including no concessions on DACA, tougher immigration measures, and more military readiness.
With the 1/19/18 appropriation deadline coming up, both the Dems and the Freedom Caucus were vying to be the partner. The Dems thought they had a deal, but it fell apart. The Freedom Caucus swooped in and made a deal for almost a clean bill--few demands at all. That bill passed the House just one day before the deadline.
The Freedom Caucus agenda is a better fit in general for the GOP. But this is complicated by the slim GOP margin in the Senate, and some very independently-minded GOP senators. Several senators, such as Lindsay Graham and Jeff Flake, want to do an immigration reform deal that is more liberal than what the rest of the GOP senators want. (This is how the Dem deal fell apart. The Dems negotiated only with Lindsay Graham, and the deal was spiked by Trump.) Do Graham and Flake have to willpower to stand strong and send the Freedom Caucus deal to defeat? I rather doubt it.
I think Graham and Flake will have to settle on a lesser immigration deal in the near future, or maybe no deal at all. Democrats have been threatening a filibuster and hinting at a government shutdown if they don't get a deal on DACA. I predict any shutdown will fail. The Dem leadership probably realizes this, so I don't think they would attempt anything more than a token shutdown at most.
Shutdown threats are a blunt instrument, and not useful for getting major changes in potential legislation. Making an advantageous deal is going to be hard for the Freedom Caucus or the Dems. The best deal may be for them have secret talks with as many allies in the GOP establishment as possible, be prepared to make concessions, and then take what they can get. However, this strategy is certain to piss off true believers among the supporters, and get the politicians labeled as sell-outs. But that's the breaks when you negotiate from a position of weakness and you aren't the only potential partner. If someone is ready to trip you and underbid you, you need to be ready to do the same.