Let me start by quoting Maimonides (Ty @Yishai) who is discussing an inadvertent transgression of a gentile of one of the Seven Noahide laws, which are punishable by human court. http://m.chabad.org/library/article_cdo/aid/1188355/jewish/Melachim-uMilchamot-Chapter-10.htm
"A gentile who inadvertently violates one of his commandments is exempt from all punishment with the exception of a person who kills inadvertently. In such an instance, the redeemer of the blood is not executed for slaying the killer, nor may the latter seek asylum in a city of refuge. However, the court will not execute him.
When does the above apply? When he inadvertently violates a command without sinful intention; for example a person who engages in relations with his colleague's wife under the impression that she is his own wife or unmarried.
If, however, one knew that she was his colleague's wife, but did not know that she was forbidden to him or it occurred to him that this act was permitted or one killed without knowing that it is forbidden to kill, he is considered close to having sinned intentionally and is executed. This is not considered as an inadvertent violation. For he should have learned the obligations incumbent upon him and did not."
In this interesting case you have presented, the gentile will be keeping the Sabbath by the type of accident which is not punishable, being that he kept the Sabbath under a false pretence of having converted, similar to "a person who engages in relations with his colleague's wife under the impression that she is his own wife or unmarried".
It should be noted that keeping the Sabbath is a heavenly offence not punishable by human courts, so this idea may not hold true.
One more idea found in commentaries dissection of Miamonides words in the aforementioned chapter, halcha #9 which would have an impact here is the following: Rambam writes "The general principle governing these matters is: They are not to be allowed to originate a new religion or create mitzvot for themselves based on their own decisions. They may either become righteous converts and accept all the mitzvot or retain their statutes without adding or detracting from them".
Some of the commentaries take this idea very literally and therefore believe that the only prohibition is when the gentile is acting this way to create a Mitzvah. The Imrei Yosher points out that a gentile who wants to convert, but has not as of yet may actually keep the Sabbath, being that he is keeping it as future Jew, not as an inventor of something new. This however is not the common practice AFAIK. Future converts are told to desecrate the Sabbath until conversion. However, this is a valid opinion upon which one can probably rely when stuck in a situation.
As a matter of practical advice though, I would point to a similar but not analogous situation brought in the footnotes of the work Mitzvos Hashem, a work dedicated to laws concerning gentiles, page 460. A person who is questionably Jewish, such as a foundling, will have to act on the Sabbath in a way to keep it as a possible Jew, or desecrate it as a possible Non-Jew. There are a number of options based on various opinions.*
1) He should perform work in the contemporary sense, but not one that is based on the 39 forbidden acts. In accordance with the latter authorities who argue with Mishna Limelech chapter 10 of Hilchos Melachim halcha 2.
2) He should do one of the forbidden acts before evening of Friday and after nightfall Saturday. In accordance with Panim Yafos in Noach who says night follows day for gentiles.
3) He should wear tzitzis in a public domain which is allowed for a Jew but would be considered carrying for a NonJew in accordance with Chassam Soffer.
4) Or he suggests doing a half of a forbidden act, for 'amounts', shiurim, were given to Jews alone.
Any of these would seemingly alleviate the issue at hand, but it is not analogous as the orthodox would not consider an unorthodox conversion as even questionable.
*This particular issue is also addressed in Minchas Chinuch towards the end of mitzvah 32. The Shabsi Frenkel mafteach also brings sources for this on the aforementioned chapter in Maimonides.