Wow, MT, you really do catch a great deal of the Harry Carney essence!I really don't think you can approach the Carney sound unless, as Merlin says, you have the right equipment, and I don't think it's remotely true to say that you get the sound you hear in your head "regardless".
My set up: a 1945 Conn 12M and an original, unaltered 1930s (?) Woodwind Co NY mouthpiece with the largest roundest chamber possible, B5, and no baffle nonsense.
Here's a full (mobile phone) video of Serious Serenade (a bit too fast, sadly) performed at St George's Guildhall, King's Lynn: http://www.harmonyinharlem.co.uk/SeriousSerenade.mp4
A brief solo after the trombone opening in this moblile phone recording of my arrangement of Sweet and Lovely: http://www.harmonyinharlem.co.uk/Olney/SweetAndLovely.mp4
There's a brief solo behind the trumpet 1min 50seconds into this audio recording of Anytime, And Day, Anywhere I made at a rehearsal a few months ago, which I'm particularly fond of: http://www.harmonyinharlem.co.uk/Anytime.mp3
I find it interesting that you're getting that bite and edge, out of a larger chamber lower baffle piece than I use, with essentially the same horn.