Why Beatty would not replace Jellicoe during the battle

Would David Beatty take command if Jellicoe were killed or disabled during this battle?

No. Vice-Admiral Cecil Burney and several other Grand Fleet officers, probably including Charles Madden, outranked Beatty in early 1915.

Later familiarity with Beatty as commander of the Grand Fleet can obscure the seniority list at the beginning of the war. In this scenario he remains well below several officers already serving with the battle fleet. Cecil Burney is the formal second-in-command, and Charles Madden probably also outranked Beatty. Multiple officers had to move before Beatty could historically receive the top command. If Jellicoe were lost in 1915, authority would pass through the existing senior chain. Beatty's later reputation does not alter that earlier command structure.