Perhaps. Interestingly the "hotel" that Fred ran in Alaska was, as was common there, a location catering in part to working girls, "servicing" the miners. Other than the Chinese exclusion act of the late 19th century, the only real restrictions on immigration the first decade of the 20th had to do with preventing sick immigrants coming in (tuberculosis, trachoma, etc), and also known criminals. Had Fred ever been arrested or had a conviction of some sort for running a house of prostitution he might have been denied re-entry in to the USA. The highly restrictive immigration law, with all sorts of quotas to ensure the "right sort", was passed in the early 1920s.
Fred C. Trump, assuming butterflies don't make him go away (Elizabeth was pregnant when the petition was denied), was born in 1905. Therefore he would be too young for WWI and almost 34 when WWII started. It is possible he could have served in the regular military prior to the end of the war, but at 40 when the war ended would almost certainly have been in the Volkssturm. Assuming he still lived in Bavaria, this would have had him facing Patton's Third Army had he seen action in the militia, whether or not this would have meant death or wounding is anyone's guess. The area where Friedrich came from (Bavaria), during the 1920s and early 30s, before Hitler came to power, was a stronghold of the NSDAP and being as Friedrich had decent finances coming back from American by the 1920s he could have middle class or better finances should he chose to get on board and support early. OTOH, like many, he could have seen his savings/investments wiped out by the war followed by the hyperinflation of the early Weimar period - a circumstance which attracted many to the "throw the rascals out, restore German glory" meme of the NSDAP. Or, like many Germans, he could have been pretty passive in the pre-1933 political turmoil in Germany.
No matter what, Friedrich simply being allowed back to Bavaria would have butterflied away the Donald, although a son born around the same time would be possible had Fred C. survived WWII.
The scenario concerning Friedrich and Fred C. outlined above is based on the historical realities of the period, and does not constitute a political statement, whether the Prince who denied the petition decides for clemency after all has a positive or negative effect on world history is left for the reader to decide.