If the original wording begins "The sun" and you insert "yellow" between those first two words, I can't imagine any theory under which adding ellipsis points (which indicate removal of one or more words from the original text) before the bracketed word "[yellow]" would be justifiable. So for the first inserted word, it seems to me,
"The [yellow] sun"
is clearly preferable to (and less misleading than)
"The ... [yellow] sun"
Your second inserted word is simply the original word shone recast in present tense. It follows that you did not delete any text to get to shines—so again (in my opinion) adding ellipsis points would be more misleading than helpful in this case. If (and it's a big if) the interpolations are appropriate at all, they are more accurately represented as
"The [yellow] sun [shines] beautifully"
In my view, however, sticking yellow (in square brackets) between The and sun is a highly dubious thing to do, unless the original text specifically noted the yellowness of the sun shortly before the quoted sentence. After all, by presenting the sentence as a quotation, you are implying that it reflects the views and priorities of the person quoted.
It wouldn't do, for example, for me to take a quotation from an enthusiastic partisan of a new president, such as this:
"The new president promises to make everyone a winner."
and add an editorial comment of my own in brackets:
"The [sociopathic] new president promises to make everyone a winner."
—because I am not merely adding an editorial comment to the original quotation; I am also attributing the sense of that comment to the original author.
As for advice from British style guides, The Oxford Guide to Style (2002) has this to say:
An editorial ellipsis—that is, one imposed upon a quotation and not part of the original—may be combined with an editorial interpolation. When this occurs, the interpolation can sometimes obviate the need for the ellipsis, where grammatically it is clear that all or some missing matter has been replaced by an editorial gloss. Take, for example, this original sentence:
The poet's Scillaes Metamorphosis, later published as Glaucus and Scilla, is the earliest of many Ovidian epyllia in the Elizabethan period.
An editor may alter the text by either adding to or deleting from it:
The poet's [Thpmas Lodge's] Scillaes Metamorphosis , later published as Glaucus and Scilla , is the earliest of many Ovidian epyllia [minor epics] in the Elizabethan period.
Scillaes Metamorphosis [by Thomas Lodge] ... is the earliest ... [minor epic] in the Elizabethan period.
As you can see, even in the version of the quotation that reflects significant cuts, there is no need for ellipsis points before a bracketed interpolation unless significant text has been dropped immediately before the bracketed language.
Thus, Oxford recommends no ellipsis points before Scillaes even though "The poet's" has been omitted from the start of the sentence, and no ellipsis points between Metamorphosis and "[by Thomas Lodge]" even though words are dropped from the original after that interpolation. The ellipsis points after "[by Thomas Lodge]" indicate the omission of the text "later published as Glaucus and Scilla," and the ellipsis points between earliest and "[minor epic]" indicate the omission of "of many"; and because "[minor epic]" stands in for "Ovidian epyllia" in the original wording, no ellipsis points are needed between "[minor epic]" and the following phrase "in the Elizabethan period."