Hey sweetie
Sorry this is a struggle for you guys. I think this is a pretty big deal and would try to find a way to discuss it without fighting (hard, I know). The way I see it, this is going to be the hardest thing you have ever done. AND the most important; you need experienced and knowledgeable support.
I definitely want someone at my birth, as a birth coach (doula) who has confidence, experience,
the time for you and your best interests in mind - but not the doctor (medical professional is necessary, yes, but not the same as a doula/birth coach/caring midwife). I want someone to look into my eyes and tell me I can do this, tell me I am doing great, rub my back when DH is holding me up... you know. And I think it is important to have someone be your voice when you are in labour and struggling and a doctor tries to push an intervention. You need someone who is able to stand up to the doctor. I have heard that many dads, especially first time dads, are not comfortable with doing that, they just want mom and baby to get through it healthy and quickly
I don't think the cost it too much. I think the stress and uncertainty you might go through without a support person is too much of a cost
A PP said that they wouldn't want a stranger at the birth, but doulas are only as much a stranger at the birth as a doctor or midwife. Care and appointments begin during pregnancy and reach to postpartum.
I would find positive doula information to share with your DH. And say, "let's just meet the doula, together" and see how he likes her. I wouldn't pressure and I wouldn't say "I'm getting this whether you like it or not" b/c it's going to be very important to have
his support for the birth! Just regularly share with him how important this is to you. And if it keeps coming down to $$ I would get productive and earn the money to pay for it. Either work an extra few shifts, or sell some things, or cut back on buying something. It can be totally affordable and will be so worth it
Good luck xox