#JavaScript (ES7), 133 bytes

_**Edit**: Fixed to match the results returned by [Jonathan Allan's code][1], which was given as a reference implementation by the OP._


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<!-- language: lang-javascript -->

    n=>[...Array(m=k=7**n)].reduce(s=>s+/^1*5?4{0,3}3?2{0,3}6?0{0,3}$/.test((--k+m).toString(7).replace(/0[62]|2[34]|4[51]/g,s=>s[1])),0)

[Try it online!](https://tio.run/##HczNisIwGIXhvVfxLYTJ1zTprzowkxavwWWIUGpaHGsiSUYQ9dprdHOezeH9666d793xEpixBz0PYjaikZzzrXPdjZzFSWySxKDiTh/@e028aDzN9kWyaut7nlbPqi0/rtv84zLjQftACGMnekYe7C64oxnJBmPjMnUxkuVyXapHKataPWq5KlQ2pu@yLBRimuM8WEcMCCh@wMAvfEcoRbgvAHprvJ00n@wYLxS@gLEmLoWBGMTFc34B "JavaScript (Node.js) – Try It Online")

###How?

We generate all numbers of \$N\$ digits in base 7 with an extra leading \$1\$ and interpret each digit as a Roman numeral symbol according to the following table:

    0: I    4: C
    1: M    5: D
    2: X    6: V
    3: L

We replace all valid subtractive pairs of the form `AB` with `B`:

<!-- language: lang-javascript -->

    s.replace(/0[62]|2[34]|4[51]/g, s => s[1]))  // in the code
    s.replace(/I[VX]|X[LC]|C[DM]/g, s => s[1]))  // with Roman symbols

**Examples:**

- `XLIXIV` becomes `LXV`
- `XIIV` becomes `XIV`, leaving a `I` that will make the next test fail
- `IC` is left unchanged, which also leaves an invalid `I` in place

We check that the remaining symbols are in the correct order and do not appear more times than they're allowed to:

<!-- language: lang-javascript -->

    /^1*5?4{0,3}3?2{0,3}6?0{0,3}$/.test(...)  // in the code
    /^M*D?C{0,3}L?X{0,3}V?I{0,3}$/.test(...)  // with Roman symbols


  [1]: https://tio.run/##jVPRatswFH2evuIuUCxjJ@BubGBwH5a0EEgIZJkJtHlQbKXVsCUjy6PZz6eSrLieS5vpQfjK59x77rlSdVRPgl@fTgcpSmCKSiVEUQMrKyEVVFLkTaYQyukBJNXf9FdVUTklNV2LknCs6LPyYwR6HYQExnP6HAImIex9HQHlTUklURT/ZZVFh2D2@yje@Y5oFjsACfaGgb156oXgzbdm3y7sPjX7dGb3pdfjmSWpaiRv08ZWwQ6CNrRREJn4XfkOdK0FoUE61Hb@hxQs1z2837uRz4/2ZJKJhiuc@XADkTUlM22NZot05IMOuVBAigK3x/N0u5jOlqNXZC9tT84dKWrq9JlG8o2GQfLBVCy4tb@Y85xljNaacN@VnNjGjdKudi/37tzYWS@BmwT2FmvGa@BmpoMK4bDkedKf3jTiTBvi2Q4@J/BN/8uhoBzXVL3FxCz4qtNCkjiTmZVP@CPFhjQg@DCGL/7HrtqDjWyom7p5DamZ/MBcnf5RPfUuvatVGwXuxeDO5FBnrihRyT80s1Q7QM@b/BasU1z7/Tdx@ead15HRIneX9m69WoawWensUQjfETIq2/qvLnWgACKXq5KMKwcMoW5K3HprlWriBUd8hNLb9Y/Vz1td2BqpO3An/QLeAx@Pxw/c6xz8D209@lUde3DlSMOf4Oy8qHXA89Dp9AI