Not on the Galloway method personally, but mile time trials are a pain. You know you're doing it right when your breathing is really challenged. Which from the sounds of it is less ideal for you. So you could consider using a different distance as a measuring stick. The 5k/10k plans on runDisney use a 800m attempt instead of a mile.
View attachment 503291
It's shorter, and has two options for hard and not hard breathing (huffing and puffing). Alternatively, you could use a longer distance as a measuring stick. Like 5ks, 10ks, or HMs as "B" races along the way. These will be less challenging to your breathing. The disadvantage to using a longer distance as a measuring stick in the midst of a training plan is that it requires more recovery from it. A mile is a relatively short race and only requires a few days recovery at most. Whereas, a HM would require a significantly longer amount of recovery (dependent on how hard you race it of course). At the end of the day, whether you use the mile, 800m, or a longer distance, it'll give you good data on which to draw conclusions.
And don't be discouraged by the results. It's entirely possible you're just not as skilled at a mile time trial as you are at a longer distance. It's uncommon in my experience helping others to see them outperform in the longer distance, but it happens. In my own experience, I ran a 39:55 10k in Jan 2018. But I'm still off on the mile equivalent (5:38) given my PR is now recently 5:42 (May 2020). My 6:12 mile time trial earlier this year suggested 21 min 5k, 43 min 10k, and 1:37 HM which I feel confident I could smash (since I ran a 1:28 HM in November). Outside of this recent training plan where I specifically trained towards the mile, I've never been "good" at the mile time trials.
As for how to run it, I usually guide people to try and run it at the 95% effort level as fast as you can from point A to point B. So for some that means continuous running, others with scheduled run/walk, and others unscheduled run/walk. I've seen all three methodologies yield the best personal outcome in the MM when we tried other methods.