Selling to the government usually entails one of three things:
1) You can sell a commodity by being a registered vendor, e.g. through the GSA. GSA registered vendors have usually agreed to prices up front, which eliminates some of the red tape. But this doesn't seem to match your situation.
2) You can find an announced opportunity. In other words, the government is telling you that it wants something. You can then write a proposal to sell it that thing. These opportunities are normally advertised in FedBizOps. They are openly competed. The catch here is that you have to be able to provide what the government tells you it wants. You can't tell it what it should buy.
3) You can respond to a SBIR, BAA, or other R&D award announcement, for instance from NASA, the NIH, the CDC, etc. This seems to fit your situation most closely. However, as with #2, these announcements are generally tailored; they are intended to produce a final product or prototype, which performs some function that the agency already wants. If they don't want what you have, you're out of luck. But unlike FedBizOps announcements, they are often more open-ended, and hence more suited to new ideas or technologies.
You can also play the long game: develop some neat tech, and then try to build support for it within government, which then leads to an opportunity like #2 or #3. This entails a lot of behind-the-scenes politicking, which is what (for instance) large defense contractors are good at. It may entail lobbying Congress, or getting your tech in front of generals, admirals, or senior agency management, and then being able to wait a few years until the next appropriation cycle (or two or three) rolls around. For obvious reasons, this is difficult for startups and small businesses to pull off, especially if you don't already have personal connections to the agency you are selling to. Successful small businesses that sell to government are often started by people who have these ties already in place when they start.