The top courts in both Ontario and Quebec have heard challenges to the Succession to the Throne Act, 2013, both upholding the Act, but for different and somewhat contradicting reasons. For reference, here is the Act's only substantive provision:
The alteration in the law touching the Succession to the Throne set out in the bill laid before the Parliament of the United Kingdom and entitled A Bill to Make succession to the Crown not depend on gender; to make provision about Royal Marriages; and for connected purposes is assented to.
I'll start with Quebec, since it provides a clear direct answer to your question. In Motard v. Attorney General of Canada, 2019 QCCA 1826 (English translation linked) the court concludes in paragraph 95 that:
[...] Canada does not have rules of succession to the throne save for the unwritten constitutional principles of symmetry and hereditary succession determined by the United Kingdom. [...]
Note that the principle of symmetry means maintaining the same monarch and/or rules of succession as the United Kingdom.
Effectively, Canada only assented to the British changes as required by the Preamble to the Statute of Westminster 1931. Since there exist no separate Canadian rules of succession, there was no effective change to Canadian law, let alone one requiring constitutional amendment.
Leave for appeal to the Supreme Court was denied.
The Ontario challenge was Teskey v. Canada (Attorney General), 2013 ONSC 5046 (lower court decision cited, the Court of Appeal summarily affirmed). It relies mainly on precedent set in O’Donohue v. Canada, 2003 ONSC 41404.
I'll outline my opinion, but unfortunately there is no direct answer to your question as both cases revolved around an attempt to invalidate the previous rules of succession on the basis that they violated the Charter by barring people who married Catholics (which the new rules also removed). As such, there was no clearly articulated reason in Teskey as to why the 2013 Act followed proper constitutional amendment procedure or whether it was required to begin with.
O'Donahue is in agreement with Motard that symmetry and rules of succession are constitutional (para. 28 & 36-37), but asserts that there exists separate Canadian and British rules in paragraph 34:
The operation of this commitment to symmetry and union of Canada under the British Crown was demonstrated by the adoption in 1937 of the Succession to the Throne Act, 1 Geo. IV, c.16. This Canadian statute effected changes to the rules of succession in Canada to assure consistency with the changes in the rules then in place in Great Britain. The changes were necessary in light of the abdication of Edward VIII in 1936. Absent this Canadian statute, the statutory change in Great Britain to account for Edward VIII’s abdication would have been contrary to Great Britain’s commitment in the Statute of Westminster. Arguably, without this statute, Edwards [sic] VIII’s abdication would not have been effective in respect of the Crown of Canada.
This results in a bit of a puzzle. If both symmetry and separate Canadian rules of succession are part of the constitution, then on a change in the British rules, Canada must either break symmetry (presumably a constitutional violation) or amend the rules of succession (presumably a constitutional amendment). This did not matter much for Edward VIII's abdication, as prior to patriation in 1982 this was all a bit moot since the Canadian constitution was effectively British, but it does matter for the 2013 Act.
Here, Teskey paragraph 13 hints at a resolution:
In the present case, the Applicant submitted in argument that Canada could and should adopt different succession rules from those which pertain in Great Britain with the possibility of recognizing a different monarch. I reject that argument on the same basis as Rouleau J. which is that this would change our present constitution in a fundamental manner and would involve the court changing, rather than protecting, our fundamental constitutional structure.
and also paragraph 15:
[...] the rules of succession and the requirement that they be the same as those of Great Britain, are necessary to the proper functioning of our constitutional monarchy [...]
I believe that the most sensible reading is that Canada was constitutionally required to adopt the 2013 Act both to maintain symmetry and to respect the constitutional Statute of Westminster 1931.
In light of this, it's my opinion that a constitutional amendment for modifying Canadian rules of succession was not required, since the constitution already foresaw this scenario.