I suggest that the short answer is yes.
In a post shared here I made the case that:
- The gifts of the Magi were very likely used to finance the trip to Egypt
- Joseph & Mary didn't yet have those funds when the avian sacrifice was made in the temple
I understand some do not like that perspective; please see the linked post for a more extended discussion of why I believe Joseph & Mary were willing to spend the money given by the Magi to Jesus.
Leviticus offered the exception not for anyone who didn't feel like offering a sheep, but specifically for those who could not afford one. There is no indication that Joseph & Mary tried to game the system--I suggest that the passage in Leviticus, and the specific reference to it by Luke, indicates that they made an avian sacrifice because they were too poor to afford a lamb.
That the family, later in life, was not considered the "elite" of their society is made plain by the condescending nature of the comment in Matthew 13:
54 And when he was come into his own country, he taught them in their
synagogue, insomuch that they were astonished, and said, Whence hath
this man this wisdom, and these mighty works?
55 Is not this the carpenter’s son? is not his mother called Mary? and
his brethren, James, and Joses, and Simon, and Judas?
The locals didn't think "one of those people" could be anyone of consequence. It doesn't sound like they are independently wealthy.
Leviticus 12 indicates that the sacrifice in the temple for a male child was to be made after 40 days of "uncleanness" and "purifying" for the mother (see verses 2-4). The words given were those that the Israelites could understand, but the principle behind them is actually a brilliant program for maternal post-partum health & protection.
This means that the trip to the temple recorded in Luke 2:22 is 40 days (or perhaps slightly more than 40 days) after Jesus' birth.
Matthew 2:11 strongly suggests that Jesus is more than 40 days old when the wise men arrive:
And when they were come into the house, they saw the young child with
Mary his mother, and fell down, and worshipped him: and when they had
opened their treasures, they presented unto him gifts; gold, and
frankincense, and myrrh.
- They're in a house now (suggestive but not probative on its own, since we don't know 100% for sure whether the manger was in a cave, dwelling, etc.)
- The wise men are able to come visit with Mary (ergo she's no longer "unclean")
- Jesus is referred to as a young child rather than an infant
Bethlehem is 6 miles from Jerusalem--it wouldn't have taken Herod all that long to realize the wise men weren't coming back. I propose then, based on the 3 points above, and the fact that there would have been but little time between the Magi's departure and the slaughter of the children by Herod, that the trip to the temple recorded in Luke 2:22 is before the visit of the Magi--the family doesn't have the money yet.
Bonus chronology for those interested
In this video series on chronology in the life of Jesus I make the argument that Jesus was about 9-10 months old when Herod died. We don't know this with certainty, but it's a decent approximation.
Wait, wasn't he 2 years old? No, I don't think so. Herod wanted to kill this supposed threat to his reign, and his violent temperament makes it clear he was willing to kill extra people to make sure he got his mark.
Herod almost certainly rounded up to two years to make sure he didn't miss. If the Magi's information suggested Jesus was a full 2 years old, Herod would have told his soldiers to kill boys up to 3 or 4 years just to be sure. Herod was an extraordinarily violent man and I wouldn't put it past him for a moment (see Josephus for a brutally honest discussion of Herod's crimes, including the murder of his favorite wife, 3 of his sons, numerous in-laws, and plenty of others who were in the wrong place when Herod was in a bad mood).