"Whole notes" are four beat notes, "whole" or expressing one whole measure if you are in a time signature whose "fractional expression" (reading a time signature like it was a regular fraction) equals one (four quarter notes or one whole note, hence the name). So, 4/4 time or 2/2 or but not 3/4 or 2/4 or 6/8 or 5/4.
"Whole" is a little unfortunate. "Compared to what?" "Compared to a semi-arbitrary but-you-have-to-start-with-some-standard of reference 4/4 measure in which One measure equals Four quarter notes in a . . . you got it, in a 4/4 measure.
A half note does not cover half a 3/4 measure. It's two beats out of a three beat measure. I'm saying this by way of analogy for why you can't (aren't supposed to, would be misunderstood) use a "whole note" for a whole five-beat measure in a song or piece which is, at least part of the way, in 5/4. (Like the chorus of "White Room" by Cream. I mean the "bolero" drums.)
In a 5/4 measure, you clever beat changing musician, you write "whole note, ligature, quarter note," to express the whole five beats of a 5/4 measure. That's if you are really holding the note or chord for "the five."
There's are no standard "fill the space, whatever size" notations in simple, everyday western musical notation from the so called common practice period of euro-anglo-western music . . . or especially standard western music notation, although in orchestral scores there is a way to say "silent throughout" with a "whole measure rest," regardless of whether the measures are 4/4, 8/4, 12/4 or more.
Don't worry, when your musicians will see "whole note, ligature, quarter note," they will know that the bendy arch connecting the whole note and quarter note are meant to be played as one five-beat note, not a separate four beat note followed by a distinct one beat note. If they are alert, they will see the 5/4 signature and expect some unusual combinations of notational values, and so will be counting, and discover that, yes, four plus one equals five, just as you intended.
Hey! I didn't even see the score until just now, and you got it exactly right! Whole note, ligature, quarter note. Like saying, "FI-I-I-I-IVE" "FI-I-I-I-IVE" etc. where each syllable or I is a beat. The beats are felt, but not expressed.
But I strongly disagree, if I may, with what was said about writing a 3-2 sometimes and a 2-3 other times. If a " 'whole-in-quotation marks " five-beat note is really just that, it should be written the same way each time. If there are "inners rhythms" then the rhythm instruments should have their syncopation expressed. If the beat changes, these cats read and play and change the beat. But a five-beat note is a five-beat noter and you will make your session musicians ask themselves, "is this five beat note different from the other? If so, how?" You don't need to have phantom rhythms expressed inside real, solid five-beat "whole plus one" notes.
You, or the rhythm notation or the rhythm players will "cue" your whole-measure guys about the inner subtleties, and if they can swing at all they will feel your swing. They will feel it, their sheet music does not need to have "special swing encoding." At least, not for the whole-plus one notes -- if that is really what they are meant to play. "Feel" comes from somewhere else.
So, yes (I belatedly read) it IS unwieldy to use ligatures for odd-valued long notes. Maybe our notation could and should better reflect that, and maybe some Indian (from India) or Iraqi (they have a ten-beat rhythm that goes 2-3 3-2 and I think is meant to drive you crazy) -- maybe some other notational systems express longer rhythmic patterns better than ours. Our system is descended from a "church" mode in which a standard, chantable long measure used something like our whole note, half note, quarter notes, and then triplets, so, like, a 12/8 pattern or something like it.
When some secular composers borrowed unusual time signatures from traditional folk dances, they found a way to make beats and measures more or less any size and shape they needed. But the "legacy" is that a whole note means four beats, not "A measure of whatever size."
To quote your post:
Can you do the same with whole notes? Will I be understood if I use whole notes for these chords, without an attached fourth note?
No, you are not supposed to do that, and no, you will be misunderstood. The "best practice" is to consistently stay with four-one ligatures, knowing that by four plus one all you really mean is five.
The awkwardness actually has one benefit. It looks unusual, and it will remind whoever is playing, "Hey, wake up, remember, this is in FI-I-I-I-IVE" ! Odd notes for odd music.