Not a Church Father, possibly not a Church writer, but good enough for Mel Gibson, Blessed Anne Catherine Emmerich had a vision that was recorded in "The Life of the Blessed Virgin Mary". Wherein it is written:
All that has so far been recorded of the blessing given to Joachim and Anna is compiled from visions and reminiscences of Catherine Emmerich during the feast of the Immaculate Conception on December 8th. She explained, however, on that day in the year 1821 that the meeting of Joachim and Anna under the Golden Gate did not occur in December but in the autumn, at the end of the Feast of Tabernacles (which lasted from the 15th to 23rd of the month Tisri, i.e. in September or October). Thus she saw Joachim building tabernacles with his shepherds before going to the Temple, and Anna receiving the promise of fruitfulness while she was praying under a tree which formed a tabernacle. In the previous year, 1820, she had, however, stated that she remembered Joachim having gone up to Jerusalem with his offerings on the occasion of a dedication festival. This cannot be the usual Jewish dedication feast in the winter (the 25th day of the month, Kislev), but must doubtless be a memorial festival of Solomon’s dedication of the Temple
Life of Mary
So yes, there was some mention of Mary's Immaculate Conception being close to the the Haunakah
And she does address the rebuilding of the temple, which did take a long time:
I never remember seeing that Herod entirely rebuilt the Temple: I only saw various alterations being made in it during his reign. Now, when Mary came to the Temple, eleven years before Christ’s birth, nothing was being built in the Temple itself, but (as always) in the outer portions of it: here the work never stopped.
At the Incarnation of Christ the Blessed Virgin was a little over fourteen years old
She also had visions of Joachim and Anne meeting at the temple well before 20 B.C. which isn't by any means wrong since they were just beefing up the temple that the Maccabees rededicated.
I was reading Bl. Anne Catherine's visions trying to find something that didn't quite jibe with the historical accounts of the rebuilding of the temple specifically, but I'm not sure there's anything that doesn't match up. She didn't however connect the Immaculate Conception to anything Herod was doing.
Except, for one thing, which might be a stretch, that the Ark of the Covenant, being missing from the temple when it was rebuilt was there all along in Mary!
Also, Jimmy Akin has some interesting research into the various years that Jesus may have been born in. Placing His birth at 2 or 3 B.C. as a more likely guess. Doing a dangerous thing and mixing the visions with the archaeology places The Immaculate Conception at 17 or 18 B.C. which is consistent.
I was thinking "Hard Hat Area", but apparently the temple wasn't entirely cut off from regular folks at the time:
The laity could not enter certain parts of the building, therefore 1000 Levites were specially trained as builders and masons, and carried out their work so efficiently and carefully that at no time was there any interruption in the sacrifices and other services.