Q. Should this verse read “the Lord is the Spirit” (ESV etc.) or “the Lord is that Spirit” (KJV)?
2 Co.3:12 Since we have such a hope, we are very bold, 13 not like Moses, who would put a veil over his face so that the Israelites might not gaze at the outcome of what was being brought to an end. 14 But their minds were hardened. For to this day, when they read the old covenant, that same veil remains unlifted, because only through Christ is it taken away. 15 Yes, to this day whenever Moses is read a veil lies over their hearts. 16 But when one  turns to the Lord, the veil is removed. 17 Now the Lord  is the Spirit, and where the Spirit of the Lord is, there is freedom. 18 And we all, with unveiled face, beholding the glory of the Lord,  are being transformed into the same image from one degree of glory to another. For this comes from the Lord who is the Spirit.
 3:16 Greek he
 3:17 Or this Lord
 3:18 Or reflecting the glory of the Lord
There is no manuscript variation in 3:17; all Greek texts read “the Lord is the spirit” ( ὁ δὲ κύριος τὸ πνεῦμά ἐστιν), but there is a “the” (τὸ) in the text, unlike “God is spirit” ( πνεῦμα ὁ θεός ) in John 4:24, (which is probably to be read simply as “God is light” “God is love”, and so on) the phrase in 3:17 does not mean “the Lord is spirit”. Nor is the “the” (τὸ) a “that” (ἐκεῖνος) so the reading “the Lord is the spirit” is more natural.
The reading of the Douay-Rheims Bible “the Lord is a spirit”, seems without any justification.
The reason for the KJV “the Lord is that Spirit” may have been to distinguish from the Trinitarian idea of a personal “Holy Ghost” or “Holy Spirit” as third person of the Trinity; since it would contradict the doctrine of the Trinity to have “the Lord [Jesus] is the [Holy] Ghost”. Or it may just have been for ease of reading, to connect 3:3 3:6 3:8 with 3:17 and 3:18:
2 Co 3:3 And you show that you are a letter from Christ delivered by us, written not with ink but with the Spirit of the living God, not on tablets of stone but on tablets of human hearts.
3:6 who has made us competent to be ministers of a new covenant, not of the letter but of the Spirit. For the letter kills, but the Spirit gives life.
3:8 will not the ministry of the Spirit have even more glory?
But if the view is taken that “spirit” is used figuratively, which is the normal Old Testament and New Testament use, then to say “the Lord is the spirit” or “that spirit” comes to much the same conclusion: Christ is “the/that” spirit in 3:3, 3:6, 3:8.
This is not saying anything more, in fact saying much less, than the parallel between 1 John 2:1 where Christ is called the “Comforter” as the “Spirit of truth” is called the “Comforter” in John ch.14, 15, 16. (see also Who or what is the Advocate or Comforter?). So ultimately the spirit, if personified, in 2Co.3 must also be Jesus himself.
Who else would it be?
2Co.3:17 Now the Lord is the/that Spirit [mentioned above 3:3-3:8], and where the Spirit of the Lord is, there is freedom.
Versions with “the Spirit” include: Wycliffe 1395, The Bishop’s Bible 1568, RV 1885, Darby 1890, ASV 1901, RSV, NASB, NIV, NRSV, ESV
Versions with “that Spirit” include: KJV 1611
Versions with “a spirit” include: Tyndale 1534 “The Lorde no dout is a sprete”, Douay-Rheims