Headhunter lowballed me even though they've been chasing me for a year

I work remote for a small company, and I've been with them for ~7 years (throughout uni and after graduation). I have a senior job title ("head"), but I am underpaid because of the company's budget. I am okay with it because of other benefits (holidays, lots of travel, 0 stress).

One of our clients (a bigger US company) has been grooming me for almost a year, trying to headhunt me behind my boss's back. Met them plenty of times, had lunch with the CEO etc, and I entertained the idea just to grasp what they would offer. I'm not opposed to changing jobs for the right offer.

When I got the offer, I was offended. When they told me what the job would entail, I already struggled with the fact the workload would at least double. But then they asked me my current salary and then offered me a mere 15% on top, with a junior job title, half the PTO and EST working hours (I'm based in Europe so that would mean afternoon into late evening). I respectfully declined (citing PTO and time zone, I didn't even want to mention compensation). They insisted and wanted to negotiate and find a solution (how on earth does one negotiate time zone). They put the numbers down and I declined again, saying the compensation and benefits would have to be substantially higher for me to consider it. Now it seems they are annoyed (they have stopped replying).

I don't understand - they have been hounding me for months, CEO, HR, CCO. And then they offer me this?! I feel offended, but now that they've ghosted me I'm starting to wonder if maybe I was in the wrong. I suppose 15% more than one's current salary is a good deal for most, but I also know I am underpaid (and yes I slightly inflated the number, but still they would be able to figure it out).

I'm not even sure what my question is. I just know it's a worrying situation as they are still our clients and I still have to deal with them, and I also fear word of this might eventually get to my boss.