There are a few issues with this idea.
First, you say that it wouldn't be that expensive at £185/year, but that's the price when you're also offering rights to others in the Union to live and work in your country and giving access to your market to the rest of the Union's goods. The £185/year would barely scratch the surface.
Second, who would the cash come from and go to? Presumably, it would be from individual Britons. Would it go to the rest of Europe? £185/year would be very cheap to gain access to the right to live and work in Europe. In fact, countries do sell citizenships, but for a lot more than that. I believe Malta is the easiest way to buy yourself EU citizenship and requires a gift of €650,000 to the government, €150,000 parked in government bonds for 5 years (probably earning next to nothing), and buying at least €350,000 in property (and holding it for at least 5 years).
Third, if you could cheaply buy the right to live and work in the EU, why would any country stay? Let me explain the scenario. If I'm a member country of the EU, I must open my borders to other EU countries, but in exchange those countries are open to me. Why wouldn't I want to simply leave and then let the minority of those in my country that want access to the rest to pay a tiny fee? In fact, if that fee is the same that I'm paying for all my citizens currently, the government could even pay it for those that want it and save loads of money. In this scenario, everyone leaves the Union because it's cheaper to just buy the right for the small few that want it. If only 1% of your population actually wants to exercise that right in a given year, you could cut your cost from £13B to £130M.
The issue is that a foreign country isn't going to give you access when your country won't give their citizens access on equal terms. It's not an issue of a small fee. Why should a British person willing to pay £185/year get access to the EU while a French person willing to pay £185/year wouldn't get access to the UK? That's the heart of the Brexit issue and why a small fee has nothing to do with the problem. Europe isn't going to give British people freedom of movement into Europe when the UK won't give Europeans freedom of movement into the UK. Why should they give UK citizens more rights than those of other European nations?
The issue is that there aren't a lot of alternatives that are in any way realistic. A realistic solution would mean that UK citizens don't get to live/work in more places than other EU citizens without some form of consideration proportional to the right. £185/year isn't proportional, especially since other EU nations must pay into EU coffers. I'm always interested in alternatives, but most alternatives sound more like, "this would be useful and cheap for me and next I'd like a free, unlimited mobile plan with no strings attached." Of course it would be useful if the EU let you pay a tiny fee for the right to live and work in the rest of the EU. Would the UK also let Europeans pay that fee to live and work in the UK? No? Ah, there's the problem.
A realistic solution could be one where a country in the EU allows UK citizens to become citizens of their country provided they renounce all ties to the UK and give up all rights in the UK. That would allow UK citizens to determine which set of rights they wanted to live under - but they'd lose UK rights if they chose the European side. But that's not "I get the best of both worlds." If you want rights in both the UK and Europe, then you need to be willing to offer those rights to EU citizens, not just UK citizens. To be fair, you might be more than willing to offer that if you were part of the remain camp, but that doesn't square with Brexit.
Ultimately, the issue is just that Brexit is a rejection of the type of thing you think could be nice. Brexit is a rejection of freedom of movement. Brexit is a rejection of making national citizenship less meaningful. Brexit is a rejection of the idea that you should be able to easily live and work where you want to. Your idea might be cool, but it's the opposite of Brexit. Maybe we should move to a world where we can freely join "nations" or "unions" that square with what we want in life rather than being citizens of where we were born or whom we were born to. But that's the opposite of the sentiment expressed by Brexit which seeks to limit that type of freedom.