As mentioned by @Steven. You'll want to get yourself an ammeter, and figure out how much current the appliance is drawing. While shopping for the meter, you'll want to look for a "Peak Hold" feature (though it might go by other names depending on the manufacturer). This feature will allow the meter to retain the highest measured value. That way you don't have to monitor the meter.

Remember when using the meter, you have to clamp it only around one of the circuit wires. If you clamp it around the cord; for example, you'll always get a reading of 0. The best way to do this, is to clamp the meter around the ungrounded (hot) conductor in the panel where it connects to the breaker.

Circuit breakers include overload protection, which is also known as thermal protection. This is typically a bimetallic strip, that pulls the contacts open if it gets too hot. This type of protection is time delayed, since the device has to physically heat up. If the current through the breaker is really high (but less than what trips the magnetic protection), the device will heat up quicker and react more quickly. In your case, however, it's more likely that the current is just high enough to cause the breaker to slowly overheat. This would explain why letting it cool down (waiting), would allow it to run longer once reset.

Your intuition could also be correct, and a loose connection could be exacerbated by the heating due to high current draw.

In either case, an ammeter will be a useful tool to start diagnosing this issue.

NOTES:

• If the breaker is a GFCI breaker, the sanitize cycle could simply be leaking too much current.
• If the breaker is an AFCI breaker, the sanitize cycle may be doing something strange that the AFCI is picking up.
• Thermal protection in circuit breakers can be affected by the ambient temperature, and the temperature inside the panel. So if the panel is in a really hot location, it could trip before it typically would/should.

Update:

Now that it's clear that the circuit is protected by an arc-fault circuit interrupter (AFCI) circuit breaker, I'd like to change my answer. While an ammeter is useful, and testing the max. current draw might be useful. I suspect that the machine is doing something during the sanitize cycle, that the AFCI doesn't like.

To test a bad neutral simply test the known hot to a good ground. Hot to ground should return nominal voltage ~ 110 - 125 Volts and hot to neutral would read something irregular.

In the case the ground is either bad or missing simply run a drop cord from a working properly grounded outlet and test hot to ground.

FYI: The above tests are to test that the branch circuit in question does indeed have a good potential of nominal voltages ~ 110 - 125 Volts, but is unable to complete the circuit from an open neutral.

Isolating Open Neutral

Isolating the open neutral can take a bit more work, but you can save yourself some time by following these steps...

• Isolate the branch circuit breaker. See if it is labeled and what portions of the house may be using it.
• Identify what is working on the branch circuit breaker and work your way upstream toward the non-working area until you can localize it to a room or similar area.
• Once you've localized the area turn off the circuit breaker.
• Next, start by inspecting the attic or other unfinished areas such as the basement or crawl space to learn the layout of the wiring. This will give you a better idea of how the electrician wired the branch circuit and help you better isolate the problem.
• If you don't have any unfinished areas then one by one inspect behind each outlet, switch box, recessed light, or similar for any signs of a lose neutral.
• After checking and reconnecting any suspicious taps you can reset the circuit breaker and test.
• Repeat as necessary.

• In the event nothing upstream works this could be indicative of a lose neutral from within the panel itself. Unfortunately, I don't think such is a DIY job and recommend only qualified electricians inspect and repair any problems inside the panel.

Alternative Step

• Purchase a 77HP-G Tone Generator
• This device goes on the source conductors ( black and white ) after the breaker has been turned off. This only works on open circuits, that is, non-continuity between conductors. A fault to ground for example will not work.
• It sends out a audio frequency on the conductors that the probe will amplify.
• The source of the problem can be isolated to some extent but using the device takes practice.

Worst Scenario

• It does happen that some junction boxes will get covered up and hidden from previous renovations and remodels. This is an unfortunate reality that makes isolating the problem quite messy. Other than opening up the walls there is no other way.
• Similar to hidden junction boxes are wire staples and nails that during the building of the house either were hammered too hard damaging the wire conductors or penetrated the wire because of accidents, negligence, and/or improper installations. Again, the only solution is to open up walls and ceilings.