Nothing to do with two classes. There are clearly just two types of "body": physical fleshly (such as those of animals, birds, humans, and so on that Paul mentions), and non-physical spirit.
Spirit beings (or persons as you seem to prefer that word) can take on a physical form and switch between the two (unless prevented from doing so by God) in order to act on earth, as shown by the angels who met Abraham, saved Lot from Sodom, wrestled with Jacob, came down to earth to have sex during Noah's day, and so on - and of course, by Jesus while in human form. But on ascension to heaven, Jesus clearly could not have taken his fleshly body there. (Luke 24:39) He was seen as physical until he disappeared into a cloud (Acts 1:9) - that does not mean his material flesh entered heaven, as other scriptures make it plain that is not possible (nor would it be necessary).
In fact, one could argue that 1 Corinthians 15:50 disproves the "two classes" belief, because Paul clearly said "flesh and blood cannot inherit the kingdom of God". But that's not what this is about, it's about whether Jesus has a fleshly "incarnate" form in heaven.
This makes it plain Jesus did not take a physical body into heaven. There was no need for Jesus to have a physical body with him in heaven to be able to present the value of his flesh and blood to his Father as at Hebrews 9:12 (he entered "by means of" his blood, not with it).
Christ took the human nature, including the body, the flesh.
That's one of the major reasons why I took issue with the quote from the catechism, not just for the sake of 'bashing'. It's simply not scriptural or logical to talk of being "incarnate" and having "glorified flesh" in heaven.
Also, nothing to do with being Michael or not - forget that for a moment. I have not mentioned that teaching, I'm personally ambivalent about it, and it's not relevant to the discussion of whether Jesus had flesh in heaven or not.
Put simply, the scriptures show
1) Jesus was created as a spirit, non-fleshly person in heaven, by the Father.
2) He was transferred to a fleshly form/body by his Father (into Mary, to grow into a fleshly human being)
3) He temporarily ceased to exist at death (had no flesh or spirit body and was not conscious as the Bible both says and implies is the case at death)
4) He was resurrected in fleshly form by his Father (hence John 2:19* and John 20:17)
3) He surrendered this fleshly human form/body at his ascension to return to his spirit, non-fleshly form/body in the spirit realm (during which he also spoke with imprisoned spirits), taking up his rightful position at the right hand of his father where he was elevated above all other spirits (except his Father, obviously).
*John 2:19
egeiró - awaken, rise up. Jesus said he will awaken or rise up. This is assumed by trinitarians to be "I will do it to myself", but can equally mean simply "I will awake", "I will rise" https://biblehub.com/greek/1453.htm
egeiro:
To arise, to stand from a prone or sleeping position. From this base meaning are several fig. extended meanings: to wake from sleep; to restore from a dead or damaged state: to heal, raise to life; to cause something to exist: raise up (give birth to) a child.
Other verses such as Matthew 9:25 show "got up", "raise up", "he arose" describing the action of getting up.