Here's an everyday puzzle that may help.
If you travel from here to there at $30$ miles per hour and back at $60$ miles per hour, what is your average speed? Instinct says it should be the average, which would be $45$ miles per hour.
But speed is (total distance)/(total time). You don't have a distance given, but you can make one up. Suppose your destination was $60$ miles away. Then it took you $2$ hours to get there and $1$ to get back. You drove $120$ miles in $3$ hours so your average speed was $40$ miles per hour.
The moral of the story is that you can't naively average averages, and a rate is an average. So be careful when you have to compute an average rate.
In your case your MRT is like the reciprocal of the speed, whose units are hours/mile. In my example those are $2$ hours per $60$ miles for the slow trip and $1$ hour per $60$ miles for the fast return. You can average those to get the average number of hours per mile. The average is $1.5$ hours per $60$ miles. The reciprocal is $60$ miles per $1.5$ hours, or $40$ miles per hour.
So this is right:
take the average of the MRTs and then take the inverse for
calculating the gastric passage rate (first way)
Edit in light of many comments and clarifications.
The important question is "what is the right way to average the MRT values?", not "why do these two methods differ?" or even "which of these two is right?"
The answer depends on what MRT actually measures. If material moves through the gut at a constant rate then your first method is correct, as discussed above. But if material leaves the gut at a rate proportional to the amount present - that is, a fraction of the amount leaves per hour - then the process is like exponential decay. I don't know a right way to compute the average rate in that case. If you have very few values to average and they are not very different then you may be able to argue that whatever results you get are essentially independent of the way you average the rates.