The one place that I had to use Jira, it was poorly managed. The manager basically used it to create a list of every project and idea he could think of and then assigned quite a lot of them to me simultaneously.
The result for me was that I was overloaded with work. And then the manager became unavailable to discuss some important blocking items for about a week.
I am not saying that Jira or project management tools can't be helpful, but it does seem to have a tendency to encourage micromanagement as well as distracting from priority items with less important tasks and deemphasizing direct communication channels.
But of course management needs some way to know what is happening or that actual work is being done even, and a way to give direction about business goals.
I believe though that if not handled correctly this change could very well interfere with development. So you are right to be worried.
Personally I would do something like this:
1. Very quitely make a backup plan for alternative employment. Just in case.
2. Come up with an alternative to Jira that A) allows managers to to see that some development is even being done and they aren't just paying people to hang out and B) gives some details about what is being worked on. This could be just email notifications when people push to GitHub.
2b. I would also set up a weekly meeting with management to discuss dev progress and business priorities face to face. And as far as capex reporting maybe you can use a Google Docs with employee hours together with a dump of github commits, perhaps grouped by user.
3. Explain that you want the dev team to focus on their work rather than checking off boxes for a process, and they want to stay focused on core tasks rather than a lot of small unimportant ones. And therefore since the current process is working you believe adding Jira will only take away time from core activities and make the team less productive in terms of actually meeting business goals.