I’m sure there are others with greater experience than I on this subject but I can offer the following as my experience in attempting to paint Scott cylinder barrels. I had the same problem as you when I had my last Scott in the 80’s. I tried Halfords (Holts?) red engine paint and like you I found it far too bright. I overcame this by toning it down with a small amount of black paint of the same type. This produced a more subdued shade not unlike the one in your second photograph and didn’t go brown as I expected. This was the best that I could achieve at the time and was unaffected by heat.
More recently I’ve retired and been able to return to Scott ownership and have been experimenting with the Magenta (what it say’s on the tin) aerosol paint that is available from Roger & Richard Moss. I was hoping to obtain a ‘metallic’ effect (which I believe was used by Scotts originally at one time?) by spraying a thin coat of Magenta over a coat of silver engine paint. Unfortunately I found it difficult to obtain the even cover necessary and the outcome was a rather unpleasant blotchy pink. However, I think this is more to do with the painter than the paint! For the time being I’ve settled for a solid coat of Magenta which looks pretty good and will return to the problem later – I’m more interested in sorting out the mechanical issues at present.
I’ve also spoken to John Underhill about this and he informed me, if I recall correctly, that the correct shade is Alazarin Crimson. Whilst this is available as an artists watercolour paint I’ve not been able to locate it in any other form. Perhaps there is a BS equivalent?
Hope this may help in some way.