The key here was to decisively defeat the Italians in North Africa in 1940/1941 before they could be properly reinforced by the DAK.
Considering the generally extremely poor morale and equipment of the Italian Army at this stage of the North African campaign, concentrating all British land assets in the Middle East/North Africa in a general advance across Italian Libya would probably have overwhelmed the Italian defenders.
The Axis, specifically the Germans, won the race to reinforce Libya.
The 155 tanks of Panzer regt 5 of the 5th Light division arriving in Tripoli by 10 March.
While 2nd AD arrived in Egypt in December 1940 it's tanks were already worn out after the 1940 invasion scare and in desperate need of refitting and adapting to the desert conditions. The division was spit on Feb 27 with an AB consisting of 52 light and 52 cruiser tanks going to Greece. What was left in Libya consisted of 29 light tanks, 25 cruiser tanks and 48 captured Italian M13s by the end of March. This was after a general reorganisation of all available and serviceable equipment, as the vast majority of 7th AD remaining tanks were totally worn out by Op Compass. These tanks were all lost in Op Sonnenblume by about 10 April and in Greece by 30 April.
The 144 tanks of Panzer regt 8 with the 15th Pz Div began moving to Libya between April 26 and May 6 and were assembled and on the line by 28 May.
On 12 May the British Tiger convoy arrived in Alexandria with 29 Light, 82 cruiser and 135 Matilda tanks, these equipped 2 ABs in the 7th AD and went into action in Op Battleaxe on 15 June. 90-100 of these tanks were lost in Op Battleaxe.
The only possible advantage the British could hope to secure was deploying all of 2nd AD in Libya, this would give it some 81 light, 77 cruiser and 48 M13 tanks against the 155 tanks of the 5th Light Division, However as stated these British tanks were worn out, not refitted prior to being deployed to Libya and suffered horrific non combat losses in action in April 1941, so they're not an option to for the conquest of Tripolitania in the face of 155 German tanks in the 5th light division.