I haven't RTFT so apologies if I've repeated anything.
I have cancer and have had to get my bloods done pre-treatment every 4 weeks for the last 18 months now. I don't like it because they can only use one arm due to lymphoedema risk and my veins are rubbish.
I am also regularly anaemic and have had a couple of transfusions recently. They do a group & save with each test now 'just in case' to save time hanging around for a cross match if it turns out I do need a transfusion the following day. It has to be done before every transfusion for the reasons stated above. It's just one extra phial while they're doing it the others anyway (but yes the form-filling & writing on the bottle takes ages, slightest mistake & it has to be rejected, but that is to protect, not inconvenience you).
Things that help me -
I go to the hospital phlebotomy department with the forms. They've only had 1 failed attempt in all that time. Usually in first go no bother. They are experts and (in my hospital at least) really kind and trained to deal with phobias/anxiety.
Emla cream on my arm about 20 mins before. Barely feel it if that's on. I use it before cannulation too.
Make sure I've eaten well & had lots to drink in advance. The one failed attempt was because I'd gone early in the day & not drunk enough.
Ask them to use a butterfly needle rather than a normal one. They're very fine & work better with dodgy veins (also hurt less as so fine).
I know it's hard. I have the worst white coat syndrome & a terrible hospital phobia, I find my treatment incredibly stressful and anxiety-inducing, even though everyone is kind. It's not rational. I do get that.
However, it would be to your advantage if you were able to widen the pool of places you are prepared to have bloods taken - this is something you need to do for your baby. Sometimes it really just is a case of hauling up those big girl pants, deep breath, head down & go. It's just a few minutes out of your life, even if they aren't very nice nothing really awful is happening to you.
I hope you can get it sorted for your sake as much as the baby's. Blood tests have to happen sometimes and building them up to be a monster in your head is just going to give you more miserable times in the future. The reality is - and I'm speaking as someone who last had bloods done yesterday and a jab done today (no transfusion - hurrah!), it's really not that bad.