This will be a difficult question to answer with solid specifics, so I'll begin by asking one of my own, and then try to provide some background.
You noted that, "A recent Y-DNA (Big Y) result (haplogroup I-FT133405) of a direct Leger male matched exactly with a Clark." Does this mean that only the (currently) deepest classified haplogroup--and you're correct in referring to it as a subclade--is identical? Or that the Big Y data is showing no non-matching SNP variants between the two test takers?
Since Ike's mother was born in 1858, it's reasonable to assume that his father was born circa 1855, give or take. There's a decent likelihood that--after roughly 3.5 generations, or call it 112 years--to the MRCA, that there may be at least one private variant in the mix for one of them, and maybe even an additional named but non-matching SNP.
But I'll back up a bit. Closing in on the end of last year, FTDNA was nearing 90,000 total Big Y tests processed. Compared to the recent past, that's a huge number. Compared to humanity's global yDNA haplotree...not so much. Meaning that a seeming correlation with only two data points may not be indicating what it might seem to be. Matching results to surnames can be at best a bit iffy especially in certain circumstances. Patronymic naming conventions is certainly one, but another was in 18th and 19th century America where unrecorded adoptions and name changes weren't uncommon.
Before spending much time researching the Clark surname, I'd personally want to see the holistic picture of the Leger descendant's yDNA matches. If a surname is in question, it's often better to look at the proportions of reported matches than just the closest match...assuming those don't align. In other words, if there are multiple matches--not just in the Big Y results but also in the 111-marker STR panel (with a GD of 4 or closer)--evaluate the testers' surnames among the top 25% or 33%. If Clark dominates, then Clark is a good bet to chase. If it isn't Clark, or if Clark is a one-off, then that has a higher probability of being an NPE or undocumented adoption somewhere along the line,
Currently at FTDNA, only two men have tested as being I-FT133405...gotta be your two test-takers. And the estimated years-before-present mean value for the appearance of that SNP is 190 YBP, so call it 1832, with a 68% confidence interval of 1761-1893. We're in the ballpark.
Its parent SNP, I-FT133587, is much older. The 68% CI for it is reported as being 398 BCE to 418 CE. That, combined with the fact that FTDNA is showing 23 additional SNPs cataloged under FT133405 would indicate to me that there are at least a few more branches waiting to be determined, waiting on additional Big Y test-takers in order to allocate the demarcations. Which would also lead me to bet that your two Big Y test takers do have non-matching SNPs.
If you want to take a deep dive into TMRCA calculations with yDNA, FTDNA is basing at least some of its results with the fairly new FamilyTreeDNA Discover tool on this research paper: McDonald, Iain. "Improved Models of Coalescence Ages of Y-DNA Haplogroups." Genes 12, no. 6 (June 2021): 862. DOI: https://doi.org/10.3390/genes12060862.
In that paper, Iain provides an example calculation which shows an average mutation rate of 83 years per identified Y-SNP. Unfortunately, a lot of folks have read that as being the average, not just one value as derived from the variables defined for that particular example calculation. That is not a number that can be applied generally across the haplotree. This can be seen empirically simply in the number of clades currently cataloged under the various basal clades. The tree at FTDNA has (today; it increases monthly; in June 2022 it was 55,989) 67,153 defined branches. Of those, 31,790 are in the R clade; that's 47.3% of all the branches. In comparison, haplogroup I has 10,133 branches, haplogroup E 5,106, and G 1,946. So clearly the same mutations-to-TMRCA can't hold across the entire haplotree.
As Iain summarizes: "For small groups, perhaps up to ~10 individuals, the dominant uncertainties are Poisson uncertainties from the small number of mutations, and the method by which mutations are counted; for larger, older groups, the accuracy of the mutation rates dominates; while for larger but recent groups, a number of second-order statistical and genetic effects become important.... The mutation rate, μSNP∼8×10−10 SNPs per base pair per year, must be definable over all b loci, so further restriction of b may be needed to account for this."
Iain's example calculation also used as a foundation that the average commercial test was obtaining about 15 million callable loci (the value of b), but the Big Y-700 is achieving a higher count than that, up to around 22 million.
The 83-year number can't be taken literally, but given a very broad view of the haplotree it, too, is in the ballpark. If we were simply to apply the 68-95-99.7 "rule" pretending that 83 years is a known median and that we have a constant, even Poisson distribution, a 68% CI would place us at 42 years to 124 years; a 95% CI would push that out to 143 years. If I'm doing a SWAG with no other information to go on, that's what I'll use: the 68% CI of 42 to 124 years.
When dealing with yDNA, the average generational interval--the average age of the parent taking into account his/her age at the time of birth of each child--is somewhat longer than the average. Generally used by Group Project administrators at FTNDA is 32 to 36 years. I usually stick with 32. So still on our SWAG, that would translate--per mutation (i.e., non-matching variant)--to 1.3 to 3.9 generations.
That's super-rough, back-of-a-napkin stuff...done at 15 minutes before last call in a noisy bar. 
But at least it's a place to start.
Circling back from yet another long and rambling post, the FT133587-positive SNP is rock solid evidence that the two men share a common patrilineal ancestor probably in line with that projected ~1850 birth. But in isolation I wouldn't lean too heavily on the assumption that Clark is the correct surname.
I'd want to make certain I included all the data I could from existing yDNA tests. In particular, I'd want to see the entire match list (it can be downloaded as a CSV file if your cousin is willing to share it with you), and I'd want to see what non-matching variants were reported between the two men. Counting those non-matches may give you a better guess for the time to most recent common ancestor.
And if there are few results to compare, I'd want to identify and recruit test-takers from both your Leger side and what you know of the Clark side. Even 37-marker STR tests could help because, if there is an NPE or undocumented adoption, that could tell you.
While there seems to be know Leger or Ledger project active at FTDNA, there is a large Clark/Clarke project: https://www.familytreedna.com/groups/clark. I would encourage your tested Leger cousin to join that project so that you could at least see if/where he is grouped and compare the STR results.
Best of luck with the search! 
