I...don't think so.
If Jakob Kupler were Adrian's father, he'd have reached out to pluck an heir for his empire--maybe a more likeable face for the control of his harem. At the very least Jakob would be tempted to use Adrian as a pawn in his struggle with his father and brother. And Rain most definitely would not be working to help Adrian against Jakob. The chronological math might add, but the rest doesn't seem to.
However, you are right. We have three possibilities
Possibility #1--
Patrick Cavanaugh is the biological father of Adrian Cavanaugh and Pearl/Miriam Cantor. This is the one we seem to be leaning toward. How does this work? Pat's son and heir to everything (including the legacy Rain is fighting to protect) is Adrian. Pat and Miel had a lot of very gorgeous sexy fun with Jacqui during "The Gig" and Jacqui got pregnant with Pearl as a result. So, Pat is Pearl's biological father. How all that led to what looks like a shotgun marriage and short-lived union between roadie/manager/groupie Jacqui and ambitious entertainment lawyer Ezra is unclear.
Possibility #2--
Ezra Cantor is the biological father of Pearl/Miriam Cantor and Adrian Cavanaugh. Doesn't seem very likely, but let's look at it. Pat and Miel have a committed relationship, but clearly they enjoy their fun. It would be shortsighted to assume that they only played with young women. Ezra is ambitious and smart, but he clearly liked fun as a young attorney hustling to build up his client roster. Pat and Miel met Jacqui during the Gig and sparks flew. It seems likely that Jacqui met Ezra during the same time period. Perhaps he hit it off with the Cavanaughs as well. And, over the course of several months, fortunate accidents happened. There are problems with this theory of events. I don't think Adrian would be heir to Rain's (or Rain's mother's) promise if he were Ezra's son. But it's possible she is mistaken about his parentage, or has decided to make concessions.
Possibility #3--
Some as yet undisclosed and very lucky gentleman is the biological father of both Adrian Cavanaugh and Pearl/Miriam Cantor after crossing paths with the Cavanaughs and Jacqui around the time of The Gig This is also possible for similar reasons to the ones outlined in the other scenarios above. However, as I said, I don't think our lucky donor could be a Kuppler brother.