Any sounds that can be expressed using the Latin alphabet will be written using the Latin alphabets.
[β] will be written B̞b̞. The diacritic beneath that B is the lowering diacritic, it is used to express, yes a lowering tone, but also that B in this sense is a bilabial approximate and not B. Much like the Spanish.
[ð] will be written D̪d̪. The diacritic is the combining bridge. It denotes dental articulation. This sound this fricative as in Or the second D in the Spanish .
[ɖ] will be written D̹d̹. That diacritic should be self-explanatory if you are familiar at all with Indic languages. It is for retroflex consonants, they occur in Sanskrit, Hindi, Chinese, and many other languages.
[ʄ] will be written as Dj, that is one letter but that should be easily understood as there is no D or J in this alphabet. sounds like the ending of the French word.
[ɠ] will be written as G. I find placing a diacritic here to show that it’s implosive a bit unnecessary, there is not a velar plosive /g/ and so there is no need to write a diacritic. However, if there happens to be a phonetic situation where one finds [ɠ] difficult to pronounce, /g/ might occur as a spoken allophone in certain dialects. But once again, I won’t place a diacritic.
The [ŋ] is a very common sound, oft written as . Part of me wants to keep that as my letter but then I would love to use a double consonant. So it will be written N̟̄. The little plus sign under N shows that this N sound is pronounced more towards the front of the vocal tract, making it more fronted than regular /n/. The line above it shows it’s nasal.
[ʃ] will be written Sh. Because H occurs in our alphabet, we must make a rule that whenever S and H occur in that ordered pair, the pronunciation will be /ʃ/ unless a soft vowel follows the H in which case they both retain their respective pronunciations.
For our retroflex t, we are placing the rounded hook diacritic beneath a T in the same manner as retroflex D. As most languages denote the [j] sound using Y, so will we.
As far as the vowels go, diacritics for vowels do not, in my opinion, reflect much of their accurate pronunciation. So I am going to assign what I think matches well.
[a] is A, [i] is I, [ɛ] is È, [ə] is E̊, [o] is O, [u] is U, [ø] is ø, and lastly [ɤ] is Ù
Naturally it follows that when any vowel follows a nasal or retroflex consonant such as m, n, or n̟̄ or d̹ and t̹ the vowels become nasal. This even applies if the vowel precedes a nasal or retroflex.
Our alphabet looks like this:
Aa Bb B̞b̞ D̪d̪ D̹d̹ Dj Èè E̊e̊ Gg Hh Ii Kk Ll Mm Nn N̟̄n̟̄ Oo Rr Sh Tt T̹t̹ Uu Ùù ∅ø Yy