You may have already seen some of this but here goes:
Your granduncle's enlistment record from the National Archives:
http://aad.archives.gov/aad/record-detail.jsp?dt=893&mtch=1&cat=all&tf=F&q=Michael+E+Cosgrove&rpp=10&pg=1&rid=649683534thdivdeathcasualties.homestead.com casualty listing (line 179):
http://34thdivdeathcasualties.homestead.com/files/CAS168.htmThis file indicates he was a member of Company K and listed as MIA. It also says he was KIA which may be an assumed result of the MIA. You might want to contact the
Iowa Gold Star Museum to see if they have any specific information about MIAs. His name is linked to the Sicly-Rome American Cemetery at Nettuno, Italy.
The American Battle Monuments Commission has a listing for COSGROVE MICHAEL E on its database for the Sicly-Rome American Cemetery. Click
this search link and past the name as it appears above and you will see the link. It indicates his name appears on the tablets of the missing. You may want to contact them for further information. Here is a link for more info on the
Sicly-Rome American Cemetery You may also want to look into
The Andrews Project, a service they offer with a photo of the headstone or commemorative inscription showing the name of those interred.
I am not sure where the 168th was on June 1st. The 133rd and 135th were renewing their attack towards Lanuvio, Italy - I am assuming the 168th was also a part of this as the push for Rome was on. The 168th was near or in Villa Crocetta on May 29th (Capt William Galt won the Medal of Honor on that day so it is very well documented).
I will see if I can find out anything more. Hope some of this was helpful, good luck.
JB