The question has been edited to specify that this refers only to the Chametz problem alone. As a result, I have deleted most of my answer. I checked with my LOR (who is a rav at Ner Israel in Baltimore) and he said that the accepted general halacha is that an Israeli in chutz la'aretz for Pesach is not allowed to buy back chametz until after all eight days of Pesach.
An Israeli who is chutz la'aretz on the Eighth day of Pesach is no more allowed to own chametz (and is subject to B'al yiraeh bal yimatzeh) than he is allowed to eat it. The question as asked has the Israeli in the (Jewish owned) pizza parlor and owning the chametz (and working with it) which as far as I am concerned causes him to violate bal yiraeh bal yimatzei. It is like the reason one must lock away chametz sold to the goy and rent him the space. Note that if the Israeli's brother owns the store, he (the brother) is in violation even if he does not own the chametz itself. The next set of posting goes into why I think this.
The Tefilot of An Israeli On The Second Day Of Yom Tov When Visiting Outside of Israel seems to imply that he would not be able to buy back chametz on the eighth day. Since this is mekach umemkar (if the Israeli brother is buying back the chametz), the action is forbidden in chutz la'aretz on that day. Additionally, he is not allowed to own chametz on that day just as a ben chutz la'aretz is not allowed to own chametz on that day. Thus, even if the rabbi in Israel buys it for him, ain shliach lidvar avaeirah. Note that I said earlier that the Israeli rabbi would not have bought the chametz for him and the Israeli brother would have had to tell him that he had chometz in chutz la'aretz. If he did not tell the rabbi that he would not be in Israel for Pesach, I do not know if the purchase would be valid. Even if someone holds that the Israeli does not have to keep the second day Yom Tov, he would be forbidden to do anything that is asur to the community in chutz la'aretz because of mar'is ayin even in private. Note that even speaking about the davening, he cannot wear tefilin where anyone can see him (kal vachomer buy back chametz or own chametz, or have chametz in his property). That is if he is not allowed to perform something that is a mitzvah for him in public, kal vachomer that he is forbidden to do something that is completely asur to the community and that he is not required to do.
The Gemara in Masechet Pesachim (52) records that Rav Safra, who lived
in Eretz Yisrael and observed just one day, spent Yom Tov one year in
a community that observed two days of Yom Tov. Out of respect for the
community, which observed a second day of Yom Tov, Rav Safra, too,
refrained from Melacha (forbidden activity) on this day, even though
for him, as a resident of Eretz Yisrael, this was not Yom Tov.
Accordingly, the Shulchan Aruch writes (Orach Chayim 496:3; listen to
audio for precise citation) that a resident of Eretz Yisrael who comes
to a community in Chutz La'aretz (the Diaspora) for Yom Tov must
abstain from Melacha on the second day of Yom Tov. The Shulchan Aruch
emphasizes that this applies even if the individual comes only for a
temporary visit and plans to return to Eretz Yisrael.
It should be noted, however, that since for this person the second day
is not actually a Yom Tov, he would recite the weekday prayers, rather
than the Yom Tov prayers. Likewise, he should wear Tefillin on this
day, as he does on weekdays, though only in private, and not in the
view of those celebrating a second day of Yom Tov. (Mishna Berura,
496:3, Seif Kattan 13)
Of course, this applies only if the individual comes to the Diaspora
for just a temporary visit. If he comes with the intent of residing
there permanently, then he is considered a full-fledged resident of
the Diaspora, and he observes a second day of Yom Tov with the special
Tefilot and without wearing Tefillin.
Summary: An Israeli resident who visits a community in the Diaspora
during Yom Tov must abstain from forbidden activity on the second day
of Yom Tov, but he recites weekday prayers and wears Tefillin.
Additionally, the daf yomi in Beitzah 9A points out that when something is forbidden because of Mar'is Ayin, it is forbidden even in a private area. This is especially true for a Torah violation. Tosafos to Kesubos 60a, Magen Avraham 301:56, Mishnah Berurah 301:56with Beur Halacha.