Hi, lots of great information given by forum members. I think @ bbrown is right you really need to get to the vet as often as you can to help make going there a good experience. I have a dog who has been very frightened of going to the vet, to the degree that I have been accused of having an untrained out of control dog, she's not, she is just frightened. For me the solution was to find a new vet practice who understands my dog, had the patience to deal with her and her anxiety of being handled and possibly hurt. My new vets are fantastic, they spend time with her and are prepared to wait for her to settle down, they are kind and calm. In fact she has just had an operation at the new vets and they were so good with her that on her last visit she actually pulled me into the front door of the practice and waited for half an hour without displaying the extreme behaviours she has had before. I don't know if this helps you, but I thought I'd let you know my experience of dealing with an anxious dog at the vets.
Some really great ideas and advice here - no need for me to add to it. It's all also very helpful to me to learn about the challenges of a Lab puppy, a task I hope to be taking on for the first time within the next few months. I really hope all goes well for you. I did want to support Mags' comment above with my own experience with my rescue. He was very fearful as a pup (paired with on leash reactivity) and had somewhat similar "unexpected" fears as your pup seems to have. Among the more random were orange trees (he was very specific), fishing poles and trash cans. It was a real struggle and I honestly had moments of real panic of what I was going to do with this dog if I couldn't bolster his confidence. Cut to the chase, but he ended up acing his Assistance Dogs International test (like service dog public access test on steroids) and being (nearly) unflappable, calm, eager to work and my amazing assistance dog and best buddy for nearly 13 years. I just wanted to share this to encourage you to not give up on your pup's potential even if it all seems rather daunting at the moment.
I'm so glad I came across this post. My 10 month old is a fearful one as well. We had two previous labs and they were amazing dogs. The new guy is so different. We've had him about 5 weeks now and he's made great progress, but we have a long way to go. He was not socialized prior to us taking him. He is so fearful to go on walks, but loves to go to the park and run on a 50 foot rope. I am working with a book called Chill Out Fido and we are on the first exercise, which is the mat. It's encouraging to read these other posts and know that others have gone or are going through this and are having success. I appreciate the sharing. Hope you are having some success since you posted this. I would love to hear an update.