After some additional research & experimentation, I have a method to make and fill a custom helium balloon. 

Background

After some initial exploration, it appeared that "foil" helium balloons are the best choice for a lifting body. The balloon envelope is made of a lightweight, low-stretch plastic and they can hold helium for a couple of days. However, after a lot of research, it was very difficult to track down the exact material that foil balloons are made from. A few resources that I found had limited information:

• https://www.ehow.com/how_7540897_create-own-foil-balloons.html
  
• https://www.quora.com/Whats-a-foil-balloon-made-from
  

There were also a few trade sites that provide some details, but some of the information was inconsistent. As far as I can tell,

• Mylar (which is biaxially oriented PET or BoPET) is/was(?) used as a substrate, but maybe polypropylene (BOPP) or nylon is more common now
• It is usually metallized with e.g. aluminium to reduce its permeability to gas 
• One of the plastics involved needs to be a thermoplastic for heat sealing

On the last point, PET appears to be a thermoset plastic, which would make it unsuitable on its own for balloons. If anyone reading this knows more about these materials, I would be very curious to learn more.

Heat Sealing

Previously, I tried using the material from an emergency blanket (which should be metallized BoPET, maybe?). The dollar store blankets appeared to be ideal, but they do not heat seal -- don't use those. I tried both a soldering iron at various temperatures and a heat sealer (i.e. a nichrome wire with a PTFE tape over it -- a future project if I ever need to make more than a few of these).

I bought a few foil balloons and cut those up. They seal well with both the heat sealer & soldering iron, so that suggested it was a material issue. For 4x the cost, MEC sells a polypropylene emergency blanket with foil on one side, so I tried that: success! The non-foil side heat seals well with both the soldering iron & heat sealer (foil to foil & foil to non-foil does NOT seal).

Some notes for heat sealing:

• Using the heat sealer makes the best seams. Adjust the heating time to ensure a good seal without too much distortion.
• Using a soldering iron with a large flat tip works, but tends to apply too much heat, which results in distortion and occasionally in some tears. Use a piece of parchment paper between the tip of the soldering iron and the plastic.

Valves

Helium balloons have a clever one-way valve system made from two layers of plastic that collapse on themselves when not held open by a nozzle/straw. [Make Things Fly] has a nice video where he deconstructs a balloon and illustrates this part.

To make a custom valve, I found that you can use the material from a ziploc freezer bag (regular ziploc bags work fine, but the freezer bag material is a bit heavier). This is also a thermoplastic, and will heat seal to the polypropylene side of the emergency blanket. Construction is a bit involved

1. Cut two strips 2-3cm wide, 15-20cm long
2. Lay the strips on top of one another (making 2 layers). Heat seal the edges of the strips from the mid-point to the end. That end will be the opening of the valve inside the balloon.
3. At the mid-point, fold back one layer of plastic so that it won't be subjected to heating. Place the other on the PP side of the balloon material where you want your valve and heat seal across the strip to fix it to the balloon envelope
4. Repeat on the other side, again folding back the opposite side so that you're just attaching the valve to the balloon envelope.
5. Cut back one layer of the valve material below the horizontal weld. The remaining layer will be a backing to guide the fill nozzle into the valve.
6. Lay the strips of valve plastic on one another at the outlet area (now there are four layers: balloon-valve-valve-balloon). Heat seal through the balloon envelope along the edge of the valve to fix the valve in place and couple it to the envelope while closing any gaps.

The pictures below might help.

Testing

To test the assembly and helium fill process, I mounted a load cell on a piece of 2020 extruded aluminium to make a test stand. The load cell is your classic Amazon special and uses an HX711 ADC on a breakout board. After calibrating the load cell with some reference weights, I checked the mass of the assembled balloon:

• mass: 5.8g
• volume (approximate): 9.5L

Based on the estimated volume, helium should give a lift force equivalent to 10.5g, for a net force of 4.7g. On the test stand, I measured 2.2g, which is a bit further off than expected (25% error in volume or lift), but could be accounted for given that the helium from the party store is "at least 80% helium." After a couple of hours, the balloon is still floating here, which is encouraging.