Edited below (see Added) to attempt to address a comment/question made/posed in response to my original answer.

Using italics instead of quotation marks could be an acceptable way to avoid the problem, but that’s not your question.

Maybe the following citations from a Quebec government site would help provide some insight:

1) On garde aussi à l’extérieur des guillemets l’article, l’adjectif possessif, l’adjectif démonstratif, la préposition ou la conjonction dont on se sert pour introduire les mots cités :

• Le premier ministre a reconnu que le gouvernement était responsable de l’« erreur ».

• Quant à son « grand projet », il n’en parle plus.

• D’après ce guide, il est nécessaire de « s’exercer au moins trois fois par semaine ».

• Si nous en croyons cette théorie, la conscience n’est qu’« un moment très fugitif ».

(I realize that your example isn’t a citation, but this could confirm that your inclination to NOT include the “l'” in the quotation marks is correct.)

2) Espacement:

L’ensemble formé par les guillemets et les mots qu’ils encadrent s’espace comme un mot ordinaire.

So taking “1)” to mean that “homme” is correct and then 2) to mean that “homme” deserves/requires its own proper spacing between anything preceding and following it, that would give

… tel l’ “homme” !

with a space between “l'” and “homme”, which would at least avoid the weirdness of that “triple apostrophe.”

(Although the example above in “1)” (l’« erreur ») seems to ignore this “rule” about spacing, at least with French “guillemets” the “triple apostrophe” issue is absent.)

ADDED to attempt to address the following comment/question:

Your logic is wrong. There is no space between l’ and an ordinary
word. Appling the rule you cite yields : l’“homme”. Have you ever seen
l’ “homme” in any publication ? – 3 hours ago

I’m probably only showing just how illogical I can be at times, but here goes anyway (in spite of the fact that I have never seen “l’ “homme”” in any publication):

From the same site cited in my answer, here is the Canadian/Quebec rule for spacing before opening English quotation marks:

Guillemet anglais ouvrant: (avant = espace) (après = rien)

I guess the logic of my original answer hinges on the meaning of the two comments at the top of the cited page:

Comment #1: Le tableau suivant donne l’espacement entre les signes de ponctuation et les mots. (original emphases)

So we must ask:

Is « L’ » considered to be a word ?

If it is, then according to my interpretation of the above Comment #1, a space before the opening quotation marks would appear to be called for.

If, however « L’homme» is considered in French to be one single word, then the « L’ » part of that single word would not be a separate word itself, and therefore the cited rule would appear to be inapplicable, and the space not required.

(Personally, although not a Francophone, I don’t think that “L’homme” is one single word, for if it were I could expect to find it listed under the “Ls” in my usually trusty “Le Robert-Micro,” which I don’t [whereas “le” and an explanation of its abbreviation to “l'” is in my “Le Robert-Micro”]).

Comment #2: Lorsqu’un signe de ponctuation, quel qu’il soit (virgule, point, etc.), est collé à un mot, il forme un tout avec ce mot, et l’ensemble ainsi formé s’espace comme un mot ordinaire.
Il en va de même des signes doubles, comme les crochets, les guillemets, les parenthèses et les tirets. (emphases added)

Here, we must ask what is meant by :

forme un tout avec ce mot, et l’ensemble ainsi formé ….

It could be argued that « le tout » and « l’ensemble ainsi formé » describes a newly formed ensemble that begins NOT with a letter (subject to liaison in the case of a vowel or non-aspirated “H” as in “homme”), but instead with the mark « “ », and that as such it should be treated as any other “ordinary word” that does NOT begin with a vowel or an aspirated “H” (i.e., a separate word requiring a space before it).

(This analysis could in fact perhaps result in meaning that the correct answer to Cl-r’s question would be “… tel l(’/e) “homme” !”, but I’m not going there! [although I have seen that in French textbooks and on French tests and quizzes])