I think the last bit is key. By doctrine the British army does not swan- in the desert they do and lose but these are not the metropolitan army trained properly.They aren't going to add a section of SP guns to an Infantry Tank Brigade, because its assumed it will be supporting an Infantry Division with its own organic artillery. As they are tied to the infantry, the guns can keep up.
They are looking at possible attachments to an Independent Armoured Brigade to make it a stand-alone unit, but this is still being discussed. In any case, until they have the tanks for it its a moot point!
The Armoured Division is the base unit, its a full division with all the supporting knick-knaks. Splitting it in two is easy, and the support doesn't have to be even if you don't want that. If an Armoured Brigade is to go swanning around without an Infantry division, it will need some extras.
As far as I can tell the Independent Armoured Brigade is an accidental formation. Its supposed to be the Armoured Bde of an Armoured Division but is then retasked but not redesignated as an Army Tank Bde and keeps the motor and supporting units of an armd Bde mostly I suspect because they are in the Med and and its hard to find a place for a motor infantry bn and silly to leave them out of the OOB. The British tend to have a problem in providing enough infantry for all the tanks they have got.
And are likely to come about in the same way = Armoured Bde for a division never deployed because priority was tank bde for the ID because without the tanks the IDs get chewed up by any enemy armoured formation and without tanks the ID cannot defeat the unsuppressed machine gun or you have to break up the ID to provide enough infantry for the amd div, so you give the ID a tank Bde ( or each corps gets a tank bde then each division the each bde habitually operates with a specific bn and you start to wonder what an armoured division is supposed to do different from a standard Division.
By Alamein 8th army has 4 amd Div and 1 Army tank Bde. But of the armoured divs, 7th AD is 2 amd 1 inf bde, 1st AD is 1 amd 1 inf, 8th AD broken up and the armoured elements placed with inf divs, 10th AD actually has 3 amd and 1 inf bde. But one of the amd bde is detached to the ID and the other two have totally different TOE ( 24th having a lot of support units, some of which are then detached to ID. 10th is a oddity as its formed from 1st Cav Div and has all sorts of attachments as previously it had been guarding most of the middle east.
However this is all Middle East Command nonesense. Feb 42 the idea is fight as BDE and all the supporting arms are dispersed to two brigade groups. And lose the Crusader battles. Then the Grown ups take charge and make the Division the main formation and proceed to win every battle thereafter. The 44 organizations btw have two Maneuvre bde the Arty concentrated and later on strip the Amd inf from the Amd Bde, pair it with the Amd recon regiment and have 3 maneuver elements and a powerful artillery/engineer etc element shared.