Here's a quick rundown of what various prominent and not-so-prominent style guides say about representing ellipsis points in quoted text.
From MLA Style Manual, second edition (1998):
But if omitting material from the original sentence or sentences leaves a quotation that appears to be a sentence or a series of sentences, you must use ellipsis points, or three spaced periods to indicate that your quotation does not completely reproduce the original. To distinguish between your ellipses and the spaced periods that sometimes appear in works, place square brackets around the ellipsis points that you add. Leave a space before the second and third periods but no space before the first or after the third. Whenever you omit words from a quotation, the resulting passage—your prose and the quotation integrated into it—should be grammatically complete and correct.
From The Chicago Manual of Style, sixteenth edition (2010):
13.48 Ellipsis defined. ... Chicago style is to indicate such omissions by the use of three spaced periods (but see 13.51 [regarding placement of a contiguous end-of-sentence period]) rather than by another device such as asterisks.
13.56 Bracketed ellipses. Especially in languages that make liberal use of suspended points, it is common practice to bracket ellipses [cross references omitted]. In an English context where both ellipses and suspended points are needed, the latter may be explained at each instance in a note (e.g., "suspension points in original"); for more than a few such instances, authors may choose instead to bracket ellipses, but only after explaining such a decision in a note, a preface, or elsewhere.
From The Oxford Guide to Style (2002):
In punctuation, an ellipsis is a series of points (...) signalling an omission. Omitted words are marked by three full points (not asterisks) printed on the line, normally separated by normal space of the line in OUP style.
8.4.1 [Omitted matter in] Prose
An original sentence or fragment may include an ellipsis, for example as points of suspension, that should not be deleted. For clarity, any later ellipsis inserted into a passage must be set in square brackets. This passage from The Importance of Being Earnest, for example, contains two ellipses:
The fact is, Lady Bracknell, I said I had lost my parents. It would be nearer the truth to say that my parents seem to have lost me ... I don't actually know who I am by birth. I was ... well, I was found.
In the following extract, the square brackets make clear which are in the original and which have been added subsequently:
The fact is, Lady Bracknell, [...] my parents seem to have lost me ... I don't actually know who I am [...]. I was ... well, I was found.
From Words into Type, third edition (1974):
Points of Ellipsis
Function. Three points, spoken of as points of ellipsis, are properly used to indicate an omission, a lapse of time, or a pause too long to be indicated by a 1-em dash; if the omission occurs at the end of a sentence a fourth point is used—the period for that sentence.
From The New York Times Manual of Style and Usage, revised edition (1999):
ellipsis. The symbol for an omission should appear rarely in the news columns because it diverts attention to the editing process. (See QUOTATIONS.) But ellipses are used freely in verbatim TEXTS AND EXCERPTS.
The ellipsis consists of three periods. Separate them with thin spaces (...), which prevent a line break from occurring between them. Use an ordinary space before and after the three dots.
To recap, the last two style guides quoted above don't mention bracketed ellipsis points at all; Chicago and Oxford say that they may be used in unusual circumstances where otherwise identical-looking suspended points are frequent or nearby; and MLA requires them.
Given the specificity and exactitude that MLA insists its adherents use in the placement of bracketed intrusions into original text, it seems clear to me that it would approve of the treatment you give your original manipulated quotation:
This intellectual rule [natural right] relative to man is accessible to reason only, and it is this law [...] [that is] instituted by the supreme Wisdom.
All four other style guides, however, seem to endorse this treatment:
This intellectual rule [natural right] relative to man is accessible to reason only, and it is this law ... [that is] instituted by the supreme Wisdom.