How frustrating! I think the heart of the issue is that hospitals and GP surgeries use different "goalposts" for blood pressure (BP).
An MRI Fusion Biopsy is an invasive procedure, often done under sedation or local anesthetic. If BP is high on the day, the hospital fears an increased risk of bleeding or a cardiovascular event (like a stroke) triggered by the stress of the needle. They often have a "hard stop" threshold, frequently around 160/100 or 180/110, to protect themselves and the patient.
GPs follow NICE guidelines, which generally don't suggest starting lifelong medication unless a patient's BP is consistently above 140/90 mmHg (in clinic) and they have other risk factors. If one is just "borderline" (e.g., 150/95), the GP might prefer "watchful waiting" or lifestyle changes rather than a pill to take long term or indefinitely.
In your case, as for many others, the "White Coat hypertension syndrome may be at work, BP is only hitting those "too high" levels because you're in a hospital, anxious a prostate biopsy (I've had too and my pressure readings at the time were higher than normal, but apparently not too high for them.
Here's a possible way forward: get your GP to assure the Urology docs that your normal BP is olay. Get a decent home BP monitor - I use one that goes on the wrist - record readings twice a day for a week (sitting clamly in your favourite chair. If those are normal (say around or under 135/85), you have grounds for claiming White Coat syndrome, and evidence that you are not as high a risk as Urology thinks.
You could ask the Urology Department's Pre-assessment Clinic to write a formal letter or email to your GP, such as "We cannot perform this clinically necessary biopsy unless the patient's BP is below [X]. Please treat the patient to this target temporarily." Apparently, GPs are much more likely to prescribe a "short-term" antihypertensive if they have a specific request from a consultant.
You can approach your GP before you have your home readings, or ask them if they will go along with the letter if you do a set of readings. I've had times when the GP wanted a week of readings and I put them in a tanle in Word or Excel.
Also, a GP or the hospital can prescribe a one-off dose of a beta-blocker (like Propranolol) to be taken an hour before the biopsy. This doesn't treat chronic high blood pressure, but it stops the physical "fight or flight" response that spikes the BP during the procedure.
Hope this helps.
Stephen