I think the reference to 82nd (West African) Division in both the FindAGrave details and the membership application to join the Burma Star Association are fairly conclusive that the latter is the man you are researching, despite the anomaly with the title of his battery. Therefore this
article on 82 (WA) Div may be helpful in filling in the picture. 42 Mortar Regiment was not formed until 1 August 1944, in West Africa, before the whole Division moved to Burma, arriving there in Dec 1944. My guess is that Lt Arthur Charles Webb remained in the Far East until VJ day and probably returned to the UK in mid 1946.
Since he would have been aged 25 when he was commissioned, I suspect that the reference to the 9th Argyll and Sutherland Highlanders meant that he first joined that unit as a soldier (possibly once he reached 18 in 1935) but almost certainly before November 1938 because that was when the 9th Battallion A&SH was transferred to the Royal Artillery and became 58th* Light Anti-Aircraft Regiment, Royal Artillery. 58th LAA Regt went to France with the BEF in 1940. While he was serving in the 58th (LAA) Regt he was probably recommended for a commission and went off for officer training in early 1942. Between his commissioning and joining 42 Mortar Regt in West Africa in 1944 he may well have served with the Royal Horse Artillery.
I can suggest two ways of going forward. Firstly you can apply to the MOD for his service record - details of
how to do that here. Ignore any comments about applying to the National Archives - that's for soldiers' records. Officers' records are not currently due to be moved to TNA. This is by far the better option if you are after personal details. The down side is that it costs £30.
Secondly if you can get to the National Archives, you could search the
war diaries of 42 Mortar Regt for the period Jan - Sep 1945 (plus
1946 if required) to get some detail about what the unit did in Burma. As an officer, it is quite likely that he will get a mention or two by name.
I wouldn't bother with Army Lists or trawling through the London Gazette unless you really have nothing better to do, as all that you will learn is when he was promoted to full lieutenant (my guess would be 18 months after commissioning) and when he left the Army - again I would guess this would have been 1946 when he was de-mobbed. If you get his service record all these details will be there anyway.
*There is a discrepancy in some of the records I have found about whether the unit was initially numbered 54 or 58. A couple of sources (eg
here) say the new regiment was 54th Light Anti-Aircraft Regiment. However I can't find anything in the Royal Artillery records for a Regiment of that name, only 58th (there is a London based 54th
Heavy Anti-Aircraft Regiment but it's definitely not the same one). Possibly a call to the
Argylls' Museum could resolve the issue.