I currently DM a party that is on the cusp of finishing Lost Mine of Phandelver and has just attained level 5. I plan to next have them begin Dungeon of the Mad Mage. I have read just enough of DotMM to have a plan for getting them from one to the other. As a modified version of the Throne of the Coronal hook, the two surviving Rockseeker brothers will find that the Forge of Spells is not fully functional because the throne from the Temple of Dumathoin in Wave Echo Cave is missing. The party wants their 10% share of the mine, but it will not be able to produce anything until the throne is found. Sister Garaele's Harper contacts will be able to track the throne to Undermountain and off the PC's go.
My question is how useful it would be to supplement the adventure as written with the adventures available from Season 8 of Adventurer's League, starting with Into the Dark as appropriate for party level (and especially if that or another 5th level AL can be a better motivation to get them from Phandalin to Waterdeep)?
'Utility' for me would come mostly from helping the party of four bridge any gaps in XP needed to be appropriately leveled for the next level of the main dungeon, especially if (1) there are times when they can find access to another level without having completed a floor, if there are 'secret sections' full of xp they might miss, or (2) if there are combats they might be inclined to avoid, and thus might be 'jumping ahead'.
Also, if there are story items needed to access the next level that could be easily-missed, it might make more sense to have them locate said item in a side-quest rather than laboriously searching the level they are on.
Lesser utility might be achieved by interesting story lines or sub-plots that help the main adventure make more sense, but the party are not strong role-players and don't worry too much about plot.
Utility could also be found in a particular magic item well-suited for a character or that would help in a particularly difficult or frustrating section of the main dungeon.
The party consists of a War domain cleric of Tempus, a Bear totem barbarian, a Raven Queen hexblade warlock, and a multiclass dragon blood sorcerer / monk who is currently 4/1 and plans to go 5/1 before starting to advance as a monk.
They are mostly hack n' slashers, enjoying combat challenges and dungeon exploration but not difficult puzzles or role play. They tend to treat NPC's as 'what's in it for me?' unless they are particular sympathetic. Fetch quests are okay so long as they are rewarded and not tedious.
Please keep your answers to 'good subjective', using your experience with both the main adventure and the specific AL quests to identify ones that would or would not work well for my needs.
Note to reviewers: I understand that product recommendation questions are off topic. I believe that I have scoped this properly by asking for an evaluation of the utility of a defined set of adventures for a stated purpose. If there is a way to better scope this question I would appreciate suggestions for edits.