I would speak to asthma UK, I don't think your surgery should be refusing to see someone with suspected asthma for tests. If you have asthma you need to know so that you can be given treatment and taught how to manage it. If necessary I would buy a peak flow meter from the chemist so you can track your peak flow yourself.
I first went to the GP about possible asthma in 1999 aged 11 and was diagnosed in 2018 aged 29!
I went in 1999 as my new PE teachers at secondary school suspected asthma. I did an asthma test and saw an improvement of 11 after taking inhalers - GP said I needed an improvement of 12 for a diagnosis so I didn't get a diagnosis.
At this point I had a strong childhood history of chest infections and 4-6 week coughs after colds. This continued in my teens although I wasn't as short of breath.
In my early 20s I started getting shorteness of breath again. This was attributed by doctors to other conditions I have like fibromyalgia and ehlers danlos. It got much worse in 2016 and I raised the prospect of asthma with several GPs who all said I couldn't be asthmatic without a wheeze. This included the same GP who had recently given me antibiotics for a chest infection and had previously seen me when I was coughing up blood streaked phlegm.
In 2018 a new GP gave me a preventer and a reliever inhaler for bronchitis and she was the first to consider asthma. She told me to try the inhalers again once the bronchitis had gone to see if they helped.
I went on honeymoon to somewhere with high pollution levels a few months later and my symptoms were awful so I resumed the inhalers which helped. It was bad enough out there that we made a note of where the hospital was just in case.
That experience prompted the GP to refer me to an asthma nurse for tests. I had an improvement of 10 rather than 12 but this time I received my diagnosis - the nurse also took my symptoms into account, as well as my history of eczema and hayfever plus the history of chest infections and coughs.
So I now finally know I have asthma and am doing well on Fostair and Montelukast, the latter of which has been a miracle drug for me and made a huge difference to my difficult to control asthma.
I would push your GP to let you see an asthma nurse OP. I would also request a trial of a preventer inhaler as well as the blue reliever inhaler.