TLDR: I think the confusion here stems from extracting these two verses from their immediate context, as well as the wider context of Ezekiel. Chapter 34 is about the restoration from Exile, and chapter 37 is a messianic prophecy - there is no single "covenant of peace", because peace (shalom) is an aspect of every biblical covenant.
חיה: The 'Living Creatures' of Ezekiel
In Ezekiel 34 there is a promise to "banish wild animals (חיה־רעה, 'living creatures of calamity') from the land". If we're to understand this, we need to pay attention to how Ezekiel's prophecies handle 'living creatures'. This is key because the whole word for 'animals' stems from them being 'living things'.
Ezekiel chapter 1 opens with reference to YHWH's "living creatures", with nine references in this first chapter alone, and some parallels later on. As the book progresses, YHWH's key theme is judgement of Judah in Exile, in order that they will know that "I am YHWH" (this appears 50 times in Ezekiel, as a rejoinder emphasising the reason for the various judgements).
What does this mean? It means that YHWH is. YHWH is watching. YHWH is listening. YHWH is on the move. YHWH is alive, and indeed is on a chariot carried by living creatures (still חיה). Sixteen times YHWH asserts that indeed these things would happen "as I live",1 and so it's fair to consider that this is one of the main themes of Ezekiel as a whole: YHWH lives.
חיה־רעה: The 'Wild Animals' of Ezekiel
This specific construction, 'wild animals/evil beasts/living creatures of calamity', is used a handful of times, as follows:
Ezekiel 5:17 - "I will send famine and wild beasts against you, and they will leave you childless..."
Ezekiel 14:15 - "If I cause wild beasts to pass through the land, and they ravage it, and it be made desolate, so that no one may pass through because of the beasts..."
Ezekiel 14:21 - "For thus says the Lord GOD: How much more when I send upon Jerusalem my four disastrous acts of judgment, sword, famine, wild beasts, and pestilence, to cut off from it man and beast!"
Ezekiel 34:25 - "I will make with them a covenant of peace and banish wild beasts from the land, so that they may dwell securely in the wilderness and sleep in the woods."
A similar term 'unclean beast' is similarly used in the following passages:
Ezekiel 29:5 - "And I will cast you out into the wilderness, you and all the fish of your streams; you shall fall on the open field, and not be brought together or gathered. To the beasts of the earth and to the birds of the heavens I give you as food."
Ezekiel 32:4 - "And I will cast you on the ground; on the open field I will fling you, and will cause all the birds of the heavens to settle on you, and I will gorge the beasts of the whole earth with you."
Ezekiel 34:28 - "They shall no more be a prey to the nations, nor shall the beasts of the land devour them. They shall dwell securely, and none shall make them afraid."
'Beasts of the field' also appears a number of times throughout chapters 31-39. These beasts are a significant element of the immediate judgement falling upon Judah during the Exile. 14:21 marks them out as one of YHWH's "four disastrous acts of judgment", intended to judge the people for their sins.
Therefore Ezekiel 34:25's promise of a 'covenant of peace' is akin to re-settling Israel in their land after the Exile and re-establishing the Mosaic Covenant (as seen in Ezra, Haggai and Zechariah), which was to give them peace (shalom) in their promised land. This is emphasised in v27:
Ezekiel 34:27 - "And the trees of the field shall yield their fruit, and the earth shall yield its increase, and they shall be secure in their land. And they shall know that I am the LORD, when I break the bars of their yoke, and deliver them from the hand of those who enslaved them."
Ezekiel 37 - the same thing, or different?
The problem with reading these two passages in parallel lies in how we understand their messianic aspects. We have two passages which are relatively close together, which both clearly state that "David" will be present (34:23,24; 37:24), so would we have any textual reason to justify considering them as separate fulfilments or separate covenants?
I see a clear progression from the former passage to the latter:
Ezekiel 34:23-24 "And I will set up over them one shepherd, my servant David, and he shall feed them: he shall feed them and be their shepherd. And I, the LORD, will be their God, and my servant David shall be prince among them. I am the LORD; I have spoken."
This is consistent with what we see in the return from Exile, when YHWH gradually restores Israel and makes a firm point of re-establishing the Davidic line through Zerubbabel, who is their shepherd but not yet king:
Haggai 2:23 - "On that day, declares the LORD of hosts, I will take you, O Zerubbabel my servant, the son of Shealtiel, declares the LORD, and make you like a signet ring, for I have chosen you, declares the LORD of hosts."
see also Zechariah 4:6-10
And then we have your second passage:
Ezekiel 37:24-25 - "My servant David shall be king over them, and they shall all have one shepherd. They shall walk in my rules and be careful to obey my statutes. They shall dwell in the land that I gave to my servant Jacob, where your fathers lived. They and their children and their children’s children shall dwell there forever, and David my servant shall be their prince forever."
Here we see a covenant that is linked to the Messiah becoming King. Depending on your eschatological understanding, this is either something that either happened during the life of Jesus (as I would argue is presented strongly in Matthew), or as a prophecy which only sees its ultimate fulfilment eschatologically.
As argued above, there is no single 'covenant of peace' in common between these two passages - the earlier passage is about the re-settlement of Israel in their land after Exile (and the cessation of their 'wild beast' judgment), and the later passage is in reference to the coming Messiah. Both times peace is proclaimed, but then again peace (shalom) was a common aspect of pretty much every covenant we see in the scriptures.
1 Ezk 5:11; 14:16,18,20; 16:48; 17:16,19; 18:3; 20:3,31,33; 33:11,27; 34:8; 35:6,11