I feel for you, that's an unfortunate situation. We are around that price point and selling into a niche industry and also got these types of offers/demands earlier in our journey. I am very thankful we declined them although they were tempting. Part of building a company is building a customer acquisition process that is repeatable and scalable and bribery doesn't qualify. If you build a product without a repeatable and scalable sales system you will fail. If anything the customer acquisition is more important than the product. This is why companies like Dropbox famously faked the product demo video, got a big waitlist of customers and then made the product. You can solve just about any problem a company has so long as it has scalable + repeatable system for profitable customer acquisition.
There are other problems with this...
1. People who actually have connections and power typically also have money, and understand the rules of investing. Respectfully, I think you're dealing with lower-level people who may not be able to deliver on their promises.
2. Once you start bribing it's never enough. People who extort bribes from you in any form will always ask for more. In my experience, this is rooted both in their greed and desire to conceal their inability to actually add value. Rather than saying... "sorry I can't actually make valuable connections" they will ask for bigger incentives.
3. It will hinder your company long term, other investors and/or clients who become aware that someone got equity for free will then be anchored at free. Be confident, you're worth something.
4. There are rules around this, blame the rules. Laugh it off with them and say unless they are accredited investors individually and/or their company wants to pay $70K for the legal it's a no go. Say this nicely and they may back off. The "shit sandwich" is a great way to deliver bad news like this... say something nice, say no, say something else nice.
5. I handle most the sales for our company and you can take heart that you are probably onto something valuable, congrats. Most people have a logic flaw I exploit as a salesperson that when they want something they want it completely. Humans are very bad at indexing desire, further if you have something unique they can't get anywhere else then you are in charge. That's great, part of what you're going to have to learn is how to get customers excited enough to try/use/buy your product without being so crazed with it they insist on owning it.
6. If you can't beat back people's desire to own a piece of your company stall. Tell them you'd love to have them as a partner but all the legal details would cost a fortune. Make it clear to them that you can't "paper" a deal of any kind without money. This might make it their idea to get you some money.
I hope some/all of that is helpful. Sorry I didn't distill it better. Others may disagree with some or all of this and they are entitled to their view. Best of luck!