As a manager, this would be a non-starter if an engineer approached me with this idea. The conversation might go like this:
Me: How's that new project I assigned to you going?
Dev: I don't have all the code needed to complete the task.
Me: Well obviously, because if we had it, I would not have asked you to devise a solution and build it.
Dev: So I'm going to build my own framework at home during my spare time, then license it back to the company to integrate with.
Me: You mean, you're going to work from home as a salaried employee and then claim that you own some of the IP for the project you are on?
Dev: Hey, it's not that bad, I'm going to license it back to the company for free.
Me: And you're going to do what with this code you write at home?
Dev: Open source it and post it on GitHub.
Me: That's very noble Dev, but did you consider how that enables competitors to catch up to us faster or that it dilutes the IP of our company?
Maybe if there was already an existing open-source project we could fork, or if you had built this thing before coming on to this project, that would be one thing.
But you are basically saying your primary focus as a salaried employee will now be split between your side project and your real job.
And did you consider that you might be tainted with inside knowledge and insight of the 3rd party service that you learned about here?
Did you consider that you might be even be competing with this 3rd party company, who's our business partner, because they might be wanting or working on the same thing?
Also, their REST API may not be public and use of their service has its own licensing issues.
Did you consider that the SDK to the 3rd party service is something we might want to license as well?
Me: Did you read your employee agreement regarding moonlighting and conflict of interest?
Me: So what are you going to do during actual work hours in the office instead of building the components you need to complete the task?
Dev: I guess I'll mock out the code around a hypothetical SDK that doesn't exist yet. But that's the cool part, when I'm done with the open-source version, it would just plug-in.
Me: That's my point, you are building a software component, based on internal requirements. Your open source code would already be tainted.
Now, here's one thing you could suggest. That the company publishes the code you write as part of its open source strategy.
You could suggest that publishing the source code as open-source by the company, could be a strategic business move. It could be used to promotes the company, promote the product, or perhaps as a means to upsell something else. One common tactic: the published code is licensed on as GNU GPL for anyone that is willing to give away their integrated code. Everyone else pays a commercial license to use it.