Learn Hardcore Malay: Saya cuba melakukan kerja sebaik mungkin untuk pelanggan wanita itu. - I try to do the work as well as possible for that female customer.

Both mean try/attempt.
In everyday Malay, cuba often behaves like a modal and is followed directly by a verb: Saya cuba melakukan... (most common).
Mencuba is fine but can sound heavier or more formal. It’s also used for trying out/testing something: mencuba baju (try on clothes).
Your sentence is most natural with cuba.

Do I need untuk after cuba (cuba untuk melakukan...)?

No. After modal-like verbs (e.g., cuba, mahu, boleh, perlu, harus), Malay usually drops untuk.
Saya cuba melakukan... is preferred.
Saya cuba untuk melakukan... isn’t wrong and appears in formal writing, but many style guides consider it less natural.

Is melakukan kerja natural, or should I use bekerja or buat kerja?

Melakukan kerja = perform/do work. Grammatical and fairly formal.
Bekerja = to work (simpler and very natural): Saya cuba bekerja sebaik mungkin...
Buat kerja = do work (colloquial).
In formal contexts you’ll also see menjalankan tugas (carry out duties).

Kerja, tugas, or pekerjaan — what’s the difference?

Kerja: general work/job tasks; everyday and neutral.
Tugas: duty/assignment/responsibility; pairs well with menjalankan tugas.
Pekerjaan: job/occupation (countable), or sounds Indonesian when used for day-to-day “work.” In standard Malay, for routine work, kerja is safer.

How does sebaik mungkin work? Do I need dengan?

se- + baik + mungkin = as well/good as possible.
Both sebaik mungkin and dengan sebaik mungkin are accepted; the version without dengan is already idiomatic.
Near-synonyms/natural alternatives: sehabis baik, sedaya upaya, sebaik-baiknya (this last one leans toward “ideally/rule-of-thumb,” not necessarily “as far as I’m capable”).

Why is itu at the end of pelanggan wanita itu?

In Malay noun phrases, the demonstrative ini/itu comes after the whole noun phrase:
- pelanggan itu = that customer
- pelanggan wanita itu = that female customer
- pelanggan wanita baharu itu = that new female customer
So the order is: head noun + modifiers + ini/itu.

Does itu mean “that” or just “the” here?

itu literally means that (far/previously mentioned), but it also commonly marks definiteness (like “the”).
Without itu, pelanggan wanita could be read as “a female customer.”
ini = this (near speaker). tersebut = the aforementioned (formal).

Do I really need to mention gender (wanita)? Is it normal?

Malay typically doesn’t mark gender unless it’s relevant. Often pelanggan itu is enough.
Use wanita (or perempuan) only if the female aspect matters (e.g., for clarity or contrast).

Wanita vs perempuan — any nuance difference?

Wanita: more formal/respectful; adult female.
Perempuan: everyday/neutral for female (girl or woman); not rude, just less formal.
Avoid betina for people (it’s for animals and can be offensive).

Should it be untuk, kepada, bagi, or demi for “for the customer”?

untuk: for/for the benefit of (most natural here).
kepada: to/towards a recipient (use when something is given/said to someone: menghantar kepada, berkata kepada).
bagi: for/for the purpose of (formal, similar to untuk in many cases).
demi: for the sake of (stronger, more emotive): demi pelanggan.

Do I need seorang before pelanggan (seorang pelanggan wanita...)?

Use seorang when you want to emphasize a single, indefinite person: untuk seorang pelanggan wanita = for a female customer.
Don’t use seorang together with itu/ini (definite): not seorang pelanggan wanita itu.

How do I express tense/aspect? Does Saya cuba mean “I am trying” or “I tried”?

Malay doesn’t mark tense on verbs. Saya cuba can mean present, past, or future depending on context.
To be explicit:
- Ongoing: Saya sedang cuba...
- Completed/past: Saya sudah/telah cuba... or add a time word (tadi/semalam).

How do I make pelanggan plural?

pelanggan by itself can be singular or plural. Context clarifies.
To mark plural explicitly:
- para pelanggan (formal, people only)
- pelanggan-pelanggan (reduplication)
- semua pelanggan (all customers)

Is there a more idiomatic way to say this in a service context?

Very natural options include:
- Saya berusaha memberikan layanan/perkhidmatan sebaik mungkin untuk/kepada pelanggan itu.
- Saya cuba bekerja sebaik mungkin untuk pelanggan itu.
These foreground the idea of “service” rather than literally “doing work.”

Can I front the beneficiary phrase (untuk pelanggan wanita itu) for emphasis?

Yes: Untuk pelanggan wanita itu, saya cuba melakukan kerja sebaik mungkin.
This is grammatical and adds topical emphasis, but the original order is more neutral.

Any pronunciation tips for tricky words here?