It sounds like the range hood duct wasn't quite airtight, and some grease has escaped into the guts of the structure it was running through. This type of IAQ problem is basically a nuisance -- it's easy enough to fix by removing the culprit grease buildup from wherever it is, although finding it may be a hassle and may require management to dispatch someone to help by getting into the wall/ceiling cavities (which is where this grease buildup most likely is, either that or inside the motor). Also, seal the range hood duct with some high-temp foil tape while you're at it, and make sure there aren't random greasy buildups behind the stove or wherever.

I wouldn't focus on IAQ monitoring -- the best approach to air quality problems isn't to monitor, lab test, or even ventilate, it's to follow your nose to the source and eradicate it. Quoting Max von Pettenkofer:

If there is a pile of manure in a space, do not try to remove the odor by ventilation. Remove the pile of manure.

For more information, I would suggest reading the work of folks who have more experience with this than I, namely BSI-017 on why IAQ problems aren't going to be fixed with fancy tests, and BSI-070 on why venting isn't an answer to everything.

You're asking a lot of questions here, but I think the core ones are why is the current system unable to keep the loft cool and what can we do about it?

Why is the current system unable to keep the loft cool?

The loft is getting hotter faster than your central air is cooling it. Because the loft is at the top of the house, heat rises to it from your lower floors. It also has a lot of exterior wall and roof surface, so it's being heated up faster than other areas. Cool air from the loft is able to rapidly flow down your open stairwell. Your thermostat and most of your air vents are downstairs, so your system runs until that area is cool but leaves your loft too warm.

For all these reasons, it's best to design a cooling system with the returns positioned at the top of the conditioned space. A well-designed system for your home would have a return near the ceiling of the loft. The return and any supply vents would need to be properly sized to the required cooling load. This could be done as its own zone, with a dedicated thermostat in the loft, or by properly balancing a single system such that cool air is distributed in a way that matches where your home is getting hotter, maintaining an even temperature.

What can we do about it?

You have a number of options, from full system replacement to minor improvements around your portable AC.

• You could redesign the central air system with proper returns and supplies to the loft. Depending on how close your current system is to the ideal, you could spend between $2000 and $20,000 on this. On the lower end we'd be talking about adjusting some dampers and adding a return; on the higher end we're getting into significant rework of ducting, possibly adding zoning controls, and possibly replacing the air handler and/or condenser with something more powerful.
• Tweaks to the current system. The biggest opportunity is probably around adding a return in the loft, but you may be able to do targeted changes to various ducts to improve balance.
• If you have dampers you may be able to rebalance enough; you can also do some basic balancing by closing various vents throughout the home. Try to direct as much air upstairs; if you get to the point that the upstairs is colder than downstairs, you've won! Just undo gradually until balanced.
• Add aftermarket booster fans to direct more air upstairs. These can help a little, but reviews are mixed. The kind that are installed into your ducts are probably better than the kind that sit in your vent registers, but are of course harder to install.
• Add a ductless mini-split air conditioning system for the loft. This would be a new system with its own thermostat just for the loft. This is a fairly quick install, as only power, refrigerant, and condensate lines would need to reach the loft (no ducts). You would need to add an outdoor condenser unit. Although the indoor unit does take some space, these units are very quiet and give you a lot of control - you can turn it on only when you are using the loft, and some units even offer heating.
• Upgrade your portable AC unit. A good portable AC should do fine in your room. The better units have a two-part vent line that you'd run outside through a window, which provides intake and exhaust for outside air to cool the condenser. This air doesn't get blown into your house, nor does cool home air get pushed outside -- the AC takes air from your room, cools it, then blows it around. Some units only have a single duct and are constantly blowing some room air outside for exhaust - these are less efficient. (Note that the two ducts are sometimes bundled as one hose - but you want a system where there are two separate streams.)
• Reduce heat gain upstairs. Insulation and air sealing can make a big difference, but so can simple things like curtains and window films (if you're getting a lot of sun).

Based on what you've said, I'd probably start with trying to close off the bathroom vent from which your loft vent runs. Then try closing some other supply vents to direct more air upstairs. Getting a HVAC expert to take a first-hand look should also be helpful, and I'd suggest getting a few opinions/quotes because there are likely many ways to address this.

Good luck!