Was the tugboat ever part of a WWII battle group?

No!

A tugboat doesn't pull, or tug, it's load; rather it pushes it with that padded front end.

Their purpose is to nudge large ships around cramped harbours due to those ships having no steerageway at such low speeds. Their very low freeboard (height of gunnels above the water) makes them completely unsuitable for open water use, even though mush of the interior is compartmentalised. As they work entirely within the confines of small harbours they have no use for large fuel tanks, and consequently an operational range measured in hours rather than days and weeks.

Further, they would be a severe liability in a naval battle due to their complete lack of armament or self defence.

Also, paradoxically, large ships are capable of much faster speeds than smaller ones, due to the cube-square law (until strength of materials fails, of course). Top speed for a tugboat is typically about 14 knots, while top speed for open water vessels is above 30 knots; more than twice as great. Top speed for a tug under load. pushing a vessel orders of magnitude heavier, will be a tiny fraction of top speed, if only because the load vessel is no longer aerodynamic when being pushed sideways.

To consider one of your examples, Midway, the as the crow flies distance to Pearl Harbor is about 850 NM; at least 84 hours travel time out for a tug at top speed, plus however long for one or more refuelings. Then several times that, at least, coming back with load. No port can be without its tugs for so long when other large vessels are constantly coming and going.

The largest air-sea operation of the war (including the Pacific but excluding D-Day as too big to be one operation) was the Malta convoy of August 1942, Operation Pedestal. The Official History (Vol 2, Notes to Map 30) says that one tug was assigned to the oiling force and one to the "escort right through to Malta". No details are given, and since two of the four cruisers in this group were sunk and the other two damaged, it's a reasonable assumption that the tug, whatever her name, returned to Gibraltar while assisting a damaged ship. (It's also worth noting that the Official History is not infallible. It credits the saving of the Ohio to "the destroyer Penn and the minesweepers Rye and Ledbury"; Ledbury was actually a Hunt-class destroyer, whose captain remembers a fourth ship helping push the invaluable tanker on her way.)

Tugs were often attached to convoys that were expected to come under attack, particularly when rescue ships were not available. The problem is that being neither merchant ships nor escorts, they were usually left out of the official reports.

During combat operations, ships that were damaged and unable to steam on their own out of the combat zone, were typically towed by other combat vessels. As an example, the heavy cruiser USS Northampton attempted to tow the carrier USS Hornet at the battle of Santa Cruz:

You need to take into account the effect of slow tugs on battle groups. If they remain together the max speed of the group is that of its slowest boat. 15 knots is darn slow by WW2, or even WW1, warship standards and would severely limit long distance deployments like in the Pacific.

Tanks on the other hand are often, at least in the initial assault phases, before deep penetration exploitation, less likely to outrun their support vehicles.

A "friendlier" environment to your concept, and more akin to your example of tank retrieval, would be near coastal operations, like in the North Sea during WW2.

Preparations for Operation Overlord, the allied invasion of Normandy began in August of 1943. The operation itself took place 6 Jun 1944 – 25 Aug 1944. Almost every boat available was thrown into the mix.

Captain Publicover, Master of the U. S. War Shipping Administration Tug Farallon, was assigned to the task of towing vital military and naval equipment to the assault areas on the coast of France. By expert seamanship and navigational skill, and in spite of cross winds and rough seas, he accomplished his difficult task in a most efficient manner. His steadfast courage in the face of enemy artillery fire, heavily mined water, and sporadic air attack, was an inspiration to his crew. The courage and devotion to duty of Captain Publicover were in keeping with the best traditions of the United States Merchant Marine.

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Perry Adams of San Carlos, California, arrived at the former summer resort, Lee-on-Solent, just across from the Isle of Wight, in March of 1944. Adams was a Purser/Pharmacist's Mate on the tug, MV Farallon, which carried a crew of 32 mariners plus 11 Naval Armed Guard.

The MV Farallon was one of 10 War Shipping Administration tugs operated by Moran Towing of New York, in company with 2 U.S. Navy tugs, 2 Dutch tugs, and several British tugs. The MV Farallon was 195 feet long, 37 feet wide, draft 15.5 feet, displacement 1,063 tons, 2,250 horsepower and could do 14 knots in light conditions

http://www.usmm.org/normandy.html

During World War II the men of the Royal Navy’s Rescue Tug Service considered themselves neither fish nor fowl.

Their expertise was essential, for ocean towage requires great skill and a cool head, but as volunteers from the Merchant Navy they had to sign what was known as the T.124T agreement, which put them under naval discipline for the duration of the war. https://www.forces.net/news/tri-service/tugs-war-untold-story-royal-navys-rescue-service