Any suggestions for 2 month-long rental in HCMC?

There are fully serviced apartments, particularly in downtown, but you will pay a premium particularly for a stay as short as 2 months.  You might consider a mid-price hotel as they may be amenable to a substantial discount over the daily rates.  You won't have a kitchen but restaurant food and even more so street food is cheap in Vietnam.  Besides, you won't be changing the sheets.  They can also handle your laundry for what will seem like a reasonable price.  No particular recommendations except the obvious negatives like the Rex, Continental or the other big names in Ben Nghe ward near the foreign consulates.  D3 should have more appropriately priced places.

I suggest you book an airbnb for one night in the area you'd like to stay and then call one of the rental agencies to help them find for you and set appointments for you to view your shortlist of  apartments.  You can find the rental agencies by google.   D1 is always ideal especially if you're working in D1.

Is the OP's target of $400-550 realistic for such a short term in D1?

Hotel is easiest route, and no deposit, no utility bill. Even 6 month leases are rare. $20 daily rate is certainly possible in 2-3 star Agoda hotels in D1/D3. And if you don't like it, you aren't locked in. Maybe you can negotiate some discount for a month. Maybe you will meet someone to share with. Did you ask your employer for ideas? They should hook you up with a local to help translate etc.
Reserve a hotel for a couple nights, check with travel agents on Bui Vien St. for other options. There are tons of hotels not listed in Agoda.

I don't know about D1 and D2, but there are plenty of private rentals in other districts with much, much lower cost than renting through an agency. A couple times in the past, I had rented a sort of studio apartment (half of an upper floor above a hair salon, private entrance, a sitting room/bedroom, and a private bathroom) for 4.5 million/m, all utilities included.

In November when we return to Saigon, we'll be in one of those private rental again until we'll find an apartment that suits us. The quote they gave to me two weeks ago was  200,000/day, with a little extra for laundry and ironing service, or more if we wish for home cooked meals (prepared separately for us.)  And yes, it'll not be a long term. They know our stay could be as short as two weeks and as long as it'll take us to find a place that we can call  home for the next few years.

Those two ideas above5 are very good solutions and enable you to have options.
Also anything less than $20/day is good.

Thanks everyone for the advice. Much appreciated

There are some studio hotel apartments in Pham Ngu Lau for about 7 Million dong , rentals month to month - if you want to be in this area, guess it depends on where your internship will be. and the commute.