I've just looked through the Combat Missions for the Dockyard. They do get grindy in the latter half, but that is normal for our Dockyard concept.
I did look over the Ship-Type Specific Missions and they all look very doable in Co-Op. That is a good improvement over the previous Dockyard.
I do agree that the Spotting Damage mission in Week 5 does stand out when compared to the Spotting Damage mission in Week 7, but @Boggzyis correct in pointing out that you can complete other, easier missions during Week 5 and continue to accrue Spotting Damage over the course of the next 2+ weeks to finish out the large mission in Week 5.
Week 7 is only 350,000 Spotting Damage, which can be done in a much shorter period of time.
My recommendation for Missions to get through Week 5 the fastest in a Co-Op setting would be:
Win 6 Battles
Get 46 "Hits to Citadel" or "Set on Fire" Ribbons
Get 95 "Torpedo Hits" or "Destroyed" Ribbons (in a Destroyer) - I'd expect around 4 or 5 per match
Get 89 "Set of Fire" or "Caused Flooding" Ribbons (in a Cruiser) - I'd expect around 4 or 5 per match
Earn 95,000 XP - This can be done in Co-Op, but also in Scenarios where Signals/Camos can be particularly effective
(Carrier Players) Cause 1,000,000 HP of damage to ships using aerial Torpedoes or by Flooding - I'd expect 60-100k a Match if you only focus on this
If you do not play Carriers well, the Battleship option may take a while. I'd recommend a Battleship that is fast and has an Engine Boost consumable like a French Battleship.