Okay,firstly.The cost does not get high while doing the Masuda Method.
People keep Pokemon in the daycare after doing the masuda method because it takes a lot of time for it to be successful and by that time the Pokemon in the daycare level up a lot and the cost is gets high.However if he or she is lucky,they might get the shiny quickly.
My advice would be keeping 2 high level Pokemon in the daycare while doing the Masuda Method so that they don't level up a lot and you can get them out without paying too much.
Source: Experience.I always keep 2 high leveled Pokemon in the day care while breeding so that I don't have to pay much
Hope I helped!