What is the proper naming of Aminoacyl-tRNAs?
Minimal Answer
If all you want to do is exactly what you have specified, on the basis of a 2013 paper on the human tRNAs identified in the 1000 genes project, the format would seem to be:

Met-tRNAMet(CAU)
or
methionyl-tRNAMet(CAU)

Notice that the anticodon is written in the 5ʹ→3ʹ direction, as in the paper, and that Met is capitalized, as recommended in this IUPAC publication of 1970 (sic).
I do have reservations about this as I have been unable to find tRNA nomenclature recommendations (not even in RNA journals), and the paper concerns genes (although a different nomenclature is used there — transfer RNA-Met (CAT) 1-1 etc.). Furthermore, my recommendation to use CAU, rather than CAT (used in the quoted paper) is only based on my experience of scientists who work on tRNA, rather than their genes would use (although it can be justified in a broader context: see below).
A variation on this is to be found in a 2017 paper:
$\mathrm{Met\text{-}tRNA^{Met}{}_{CAU}}$
So I suspect there is no ‘official’ nomenclature.
Problems in giving a more comprehensive answer
The answer above is adequate in certain circumstances, e.g. in differentiating in a paper or a lecture between two species (e.g. with different anticodons, although this does not occur with methionine). The problems occur when you wish to distinguish one tRNA from all other tRNAs present in a cell or encoded in a genome. I list some potential ambiguities:

Initiating v. elongating Met-tRNAs (in the case cited)
Anticodons containing modified bases
Distinct tRNAs with the same amino acid specificity and the same anticodon
Cytoplasmic v. mitochondrial tRNAs in eukaryotes
tRNAs from different species

Possible solutions
Different Met-tRNAs
This is often most important as both initiating and elongating Met-tRNAs have the same anticodon but are structurally and functionally distinct. Differentiation is generally done by way of a subscript: originally ‘f’ or ‘m’, respectively, but for eukaryotes now generally ‘i’ or no subscript:
$\mathrm{Met\text{-}tRNA^{met}_f} \quad or \quad \mathrm{Met\text{-}tRNA^{met}_m}$
$\mathrm{Met\text{-}tRNA^{met}_i} \quad or \quad \mathrm{Met\text{-}tRNA^{met}}$
Anticodons containing modified bases
It was observed early on that the anticodons of tRNA contained modified bases — e.g. inosine, which had particular patterns of ‘wobble’ codon recognition (see my answer to another SE question). Therefore, in many cases it would be important to specifiy these — which is why I recommend using U, rather than T, e.g.

Tyr-tRNATyr(GΨA)

tRNAs with same anticodon
Gene duplication often results in tRNAs with the same specificity and the same anticodon. In some cases the sequences are identical, so there is no way of distinguishing them, but in other cases mutations will occur that do not change the function. To distinguish such species with a single base (here at position 40) the 1000 genomes paper uses the following style:

(G40) tRNAArg(GCA)
or
(C40) tRNAArg(GCA)

At the genetic level arbitrary numbering is used for human tRNAs, e.g.

transfer RNA-Met (CAT) 1-1
transfer RNA-Met (CAT) 1-2

whereas in Drosophila a variant of this style is used, e.g.

transfer RNA:Methionine-CAT 1-1
transfer RNA:Methionine-CAT 1-2

Cytoplasmic v. mitochondrial tRNAs
It is important to remember that the mitochondria of eukaryotes contain their own set of tRNAs, and to distinguish these if necessary by a preceeding ‘mitochondrial’. In Drosophila the symbol would be ‘mt:tRNA:Met-CAT’, and the name ‘mitochondrial transfer RNA:Methionine-CAT’.
Species
Obviously, one can specify the species by a preceeding adjective. I mention this to point out that set of tRNAs is different for different species.