The gemara initially suggests that the case of ופושעים יכשלו בם is a person who eats the korban pesach לשם אכילה גסה. Depending on how we interpret the phrase "l'sheim achila gassa", the pertinent sugya will either be a lishmah/kavana issue, or will be a issue related to the appetite that is appropriate for eating kodshim. I raised this gemara in my sefer nasiach b'chukecha (pg. 178) and in the hashmatos to pg. 136 as it relates to the כוונה-לשמה issue. And in the hashmatos to pg. 93 how it relates to hiddur mitzvah and eating l'teiavon).
Basically, Rashi understands that the issue is a kavana issue. The idea of eating it achila gassa is to the exclusion of eating it l'sheim mitzvah. According to this approach the issue at hand is a lack of intent for the mitzvah, thereby leading the turei even to ask a contradiction in reish lakish. Why would reish lakish consider this a case of not fulfilling the mitzah in it's ideal state, yet fulfilling the fundamental mitzvah - Reish lakish holds that mitzvos require kavana and without intent he wouldn't be fulfilling any mitzvah at all. This question is addressed by the Beis Halevi (3:51:3). It seems that the language of the gemara לשם אכילה גסה rather than saying אכילה גסה is what compels rashi to explain the problem to be his intent rather than his action.
However, the Tosafos HaRosh argues on Rashi by asking that if the problem was merely his intent, then the gemara should say that he ate it שלא לשם מצוה and not focus on the achila gasa aspect? Therefore the Tosafos HaRosh explains that the issue was not what he was thinking but rather what he did. Generally one should be eating kodshim when he is not starving to fulfill the concept of למשחה כדרך שהמלכים אוכלים. It is more respectful for kodshim to be eaten when one is not starving, but one should also be eating it while he still has a desire for food. Eating kodshim when he is already full - לשם אכילה גסה is not an ideal way to fulfill the mitzvah. The Rashash on his own takes a similar approach to the gemara. See also the Maharatz Chiyus who explains exactly the opposite approach, that he should have eaten in while not starving, but he violated the ideal mitzvah by eating it while starving to break his starvation (this approach is very difficult to fit into the language of לשם אכילה גסה).
From the Tosafos HaRosh we learn a fascinating point regarding hidur mitzvah. The more one has a desire to do the mitzvah, even if the desire is motivated by physical desires (i.e. being hungry), and not by the fulfillment of the mitzvah, it would still qualify as hidur mitzvah. Therefore, one who eats kodshim after he is totally full and has no desire to eat any more and is only eating to fulfill the mitzvah, is not fulfilling hidur mitzvah.