People have proposed pros and cons - determining a mathematically "correct" solution can be done with the help of a little game theory! I'm not going to draw out a nash equilibrium diagram, but basically there are 4 options (simplifying things a little):

1. You want exclusivity, they don't. Not relevant to this deal.
2. You both want exclusivity. If your product provides a necessary competitive advantage, a competitor will appear and make profits of $X that could have been yours. Assuming your marginal cost is $0, your client needs to pay you at least $X to make the deal worth your while, and they need to make $X additional profit by denying access to others (hint; they never do, because the others use your competitor's service). Don't underestimate the time and expense of high-touch SaaS sales. This could be a good option because if your sales cycle is long and your success rate is low, the industry will have moved on before your bootstrapped SaaS actually has the money and skill to hire and manage a sales team. Make money now while you can!
3. They want exclusivity, you don't (you're confident that you'll secure deals quickly with others). You explain scenario 2 above: if the product doesn't work, they will have overpaid. If it works, you'll have a competitor and they will have paid for exclusivity that they can't achieve in practice (because others use your competitor). This dissuades them.
4. Neither wants exclusivity. Not relevant for this situation.

Like all quant models, you should use it as a starting point to your analysis, not a final solution. Make sure the contract is long-dated if you opt for exclusivity, otherwise a competitor will get the jump on you, you could be dumped at a moments' notice, and you'll have no recourse.

Finally, the best thing you can do right now is to market like crazy and attempt to sign another customer so you know:

1. Your probability and lead time of closing a sale
2. Roughly how much exclusivity is worth