However they have different tones (as does all the words in my list).
Not quite.
khao (enter) is เข้า. Falling with a short vowel.
khao (rice) ข้าว is ข้าว. Falling with a long vowel.
khao (mountain) is เขา. Rising with a short vowel.
khao (he/she) is เขา or เค้า. Either rising with a short vowel or, more normally, high with a short vowel.
khao (white) is ขาว. Rising with a long vowel.
I don't think you can get a full house with khao but you could have had news (ข่าว) to at least get a low tone.
And it's an unfortunate choice of words because you can give fully accurate IPA transcriptions of these which only use standard Roman letters. They will all be either [kʰaw] or [kʰaːw] (with the relevant IPA tone marks).
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^ Or you could use numbers, as in pinyin.