Should one compromise on quality
That will give you the effect of spilling coffee on your pants (I shall not use the normal expression) it will make you nice and warm for a short time, but be awfully cold soon after. You will make a few quick sales and get the reputation that will make it impossible for you to make ANY sales of ANYTHING later.

Of course the easy way is to tell the customer that solution is not feasible.
Sometimes that is the right thing to do

But is there another way
Hunt, hunt, hunt and you may find something of quality that can fit the budget.

Erik

PS if your question is "how do i make a few units and sell them until I can buy in volume and make a profit at the market price" the answer is take the loss and make up for it later.

I have a board made that cost me $70 which I am redesigning to get the cost down. My prototypes of the lower cost unit cost $450.