Without more information it's not easy to answer your question, but I'll give it a fighting try :)

The health insurance industry is a commodity industry, they sell the same things at around the same prices and every little competitive edge is something they hang on to dearly. It's not surprising or even unusual for them to wish for exclusive access to your product (whatever it might be) during the pilot period which I assume would then roll into exclusivity if they roll out into a full implementation too.

That said, for your business this is a huge risk, it's the proverbial basket in which you put all your eggs and that's a significant risk to a young business, because if that pilot fails, you're 6 months behind in your goals and that's a big set back for a small company.

You can do a few things to try to minimize the risk, or maximize the reward from agreeing to their terms (because if they are a major health insurer, you'll eventually capitulate either way).

1. Limit the scope of the exclusivity - If they are concerned about exclusivity in your state, be very clear that it is limited to your state and in other states you're allowed to pursue other clients.
2. Ask them to pay for the privilege - It's not unusual for a company to offer exclusivity in exchange for recompense. You basically tell them, listen that's fine but you're asking me to limit the profitability of my company, I'm happy to give you exclusive rights to the product but the pricing model for that is different.
3. Use exclusivity as a negotiating tool - You can add a condition to the exclusivity that states that if after 3 months they haven't declared their intent to proceed to a full implementation the exclusivity is revoked.
4. Get something in exchange - You may be able to give them exclusivity on the condition that they become a referral customer and give you a case study. They'll be happy to say nice things about you whenever anyone asks if that's all it costs for them to get exclusivity, and for you having someone important at a major health insurer say your product is great might be just what you need to get 2 or 3 more pilots in different states or industries

If you're a start up and this is your first big enterprise client, they are going to try to bully you and push you around in the negotiation, they're the big kid in the school yard and you're the runt of the litter and they know it. So look for the way that you can give them what they want, but leverage it to make them more sympathetic to your situation and to invest them in your goals.

Most importantly though, make sure that you don't give up your company's ability to grow in the future by signing away something that sounds so innocent. If giving this client exclusivity is going to lock you into a position where you have to pass on other opportunities as a result, maybe they're just not worth it.

There's nothing wrong with an exclusive, it gives you an opportunity to execute well and align a strong public announcement with a high-profile client, if picked correctly.

However, six months is a little long. Typically, an exclusive like this is more along the lines of three months than six. In any negotiation, there's a side with more leverage, which I'm not able to determine with the limited information that I have. If it's you, I would pick what means more to you: A six month exclusive with more revenue or a 3 month exclusive with less and negotiate to that point.

I work quite a bit with healthcare and also have clients who actively sell software into the healthcare industry both to carriers, providers, and even government orgs. I would be inclined to provide them contract options with three and six month exclusivity terms, and then even provide a long term rate that makes it worth your while to maintain exclusivity in that territory with specific terms you deem reasonable so you can push the product elsewhere. You may request in turn for the exclusivity period, different terms so it is worth your while to provide the exclusivity. Perhaps they provide reference to any other sales opportunities outside the territory, pay increased rates per longer term they request the exclusivity, agree to let you use their band in advertising in the future pending they stay a customer, etc.

This is just a negotiation point, they have come out in want of exclusivity, figure out exactly what it is what you want out of the opportunity and start the negotiating points there.
Good luck!

Yes. Six months is an incredibly short amount of time in the health insurance industry. Most of their contracts are a minimum of a year with purchasers if not longer and multiyear. The six month would be a great time for you to gain experience and working out any issues. It could also creat and element of FOMO with their competitors. I work for a health insurance plan and typically don't ask for any type of exclusivity when we use vendors. If anything, I want others using the service so our provider networks and populations are not pushed into using a very narrow solution. I'd love to learn more about your solution and help more.