You are right. It is a breath of fresh air for me. I'm going to take this advice.

It is just feels awful to lose business that you can execute perfectly well for a reason such as "being too small".

Of course it hurts to lose any business just because some of your potential customers think your company is too small.

Serve the customers you can get now well. Not only does “many a mickle makes a muckle” (a Scottish saying that reflects pretty well a saying we have in Swiss-German) but also the concentration of risk is much lower.

And don’t forget: most companies/customers are (too) small by themselves and many of them prefer suppliers that make them feel more like VIPs than like numbers (we don’t care about you - you’re too small for us).

It’s not only a disadvantage to be small. And after you have proven that you have come to stay, bigger companies will notice it one day.

Thanks for the advice. How do you suggest I get out of this jam?

At minimum, since you're in a hole, stop digging. If you've got fake names on your website, I second makecheck's advice to use "support@", "sales@", etc. instead. Transitioning out of fake names with customers you have an established relationship with may be a little more difficult.