Note you may be able to squeeze the design to smaller area on 4-layers, making the PCB cost difference somewhat smaller.
For example, a tightly packed, miniatyrized 50x100mm 4L design might become 60x120mm on 2L due to added ground and power traces which need to be done 100% on the component layers, preventing component placement.
Cost difference of bare PCB being 5x cheaper is of little importance, if the PCB is like 5-10% of the total cost. (Besides, I think you won't reliably find a 5x price difference, unless you deliberately shop for some special discount 2L and compare it to standard 4L.)
For a quick approximation, the PCBWay calculator shows $0.85/pcs for a 4L 50x100mm, and $0.65/pcs for a 2L 60x120mm, @ 1000pcs. I would be happy to pay $0.20 extra for shorter and less painful PCB routing time, smaller final product, and better EMI performance.
I would guesstimate your component BOM would be around $10 at minimum, and the assembly cost, which would be the same for 2L or 4L PCB, would be somewhere around $1 PCB.
Yes, you may be able to find a somewhat better deal on a 2L board, but you might find a good deal on 4L, as well.
If this is just for a design challenge, then I understand.