Yes; the Silver Stag has been in circulation since before Robert's Rebellion.
Chronology
Robert's Rebellion (RR) also known as the "War of the Usurper" occurred at about 282 - 283 AC.
We see the Silver Stag featured in the Dunk and Egg Novellas (which take place roughly 89 years prior to RR in 209 AC):
The Hedge Knight
The cloth purse contained three silver stags, nineteen copper pennies
...
“Eight hundred stags, for I’m feeling kindly.”
-The Hedge Knight - A Tale of the Seven Kingdoms.
The Sworn Sword
I could pay the man a silver stag, and three to the woman for the insult.
...
“One silver stag. And three for you, m’lady.”
-The Sworn Sword - A Tale of the Seven Kingdoms.
The Mystery Knight
"We have no coin for beds."
"We have twenty-two pennies, three stars, one stag, and that old chipped garnet, ser."
...
He gestured at the piles of silver stags and golden dragons on the table.
...
He picked up a silver stag and set it to spinning with a flick of his long fingers.
-The Mystery Knight - A Tale of the Seven Kingdoms.
Too Much Work, too Little Time
If Robert was to change the sigil on the golden dragons, he would have twice the work ahead of him. If he were to change the Dragon to a stag then he would have to also change the stag to something else. Otherwise he would have to change the dragon to something else:
All of which represent some other house, so why not leave dragons which are thought to be the most fearsome, powerful and majestic creatures to represent the highest value coin?
You are also basing your premise on the fact that Bobby hated Targaryens when the only evidence is that he hated Rhaegar for "kidnapping" Lyanna.
Ah, but coins do change
You should bear in mind also that the designs of coins in Westeros do actually change; mainly on the "other side" - the "head side" - of the coin. The change happens much the same way in real life to signify the change in Monarch. But the original coins remain in circulation until naturally phased out.
In The Mystery Knight we see
Dunk speaking to Uthor "The Snail" Underleaf whilst The Snail counts his winnings from the Whitewalls Tourney. They discuss a certain "King" on one of the gold dragon coins and realise that it is the "wrong" king. Here's how Dunk outlines the previous kings on the dragons:
Frowning, he hefted the coin in his palm, examined both sides, tasted it. "Gold, not shaved or clipped. The weight feels right. I'd have taken it too, m'lord. What's wrong with it?"
"The king."
Dunk took a closer look. The face on the coin was young, clean-shaved, handsome. King Aerys was bearded on his coins, the same as old King Aegon. King Daeron, who'd come between them, had been clean-shaved, but this wasn't him. The coin did not appear worn enough to be from before Aegon the Unworthy. Dunk scowled at the word beneath the head. Six letters. They looked the same as he had seen on other dragons. DAERON, the letters read, but Dunk knew the face of Daeron the Good, and this wasn't him. When he looked again, he saw that something odd about the shape of the fourth letter, it wasn't ... "Daemon," he blurted out. "It says Daemon. There never was any King Daemon, though, only--"
"--the Pretender. Daemon Blackfyre struck his own coinage during his rebellion."
-The Mystery Knight - A Tale of the Seven Kingdoms.
You can check out the known variations in the designs of the coinage here:
Show me the Money
Some examples of different "Kings" on the head-side of the coins:
Original here.
Original here.
Other Things of Note
There are two ages of Westerosi coinage: pre and post Conquest.
The current coinage system used is from the post-conquest era. There have been known to have been at least one other type of coinage used in the pre-conquest era:
There are older coins, still in use from before the War of Conquest. In A Feast for Crows we learn of gold coins of the Kingdom of the Reach, which were known as 'hands', they feature the hand-shaped sigil of House Gardener on one side and the face of a king on the other, with each coin roughly half the value of a golden dragon.
-http://awoiaf.westeros.org/index.php/Currency#cite_note-Raffc7.7B.7B.7B3.7D.7D.7D-2.
And Finally, So Spake Martin
Yes indeed. And much more medieval. It occurs to me too that Westerosi coinage is probably more complex than actual British medieval coinage, since the Seven Kingdoms were actually seven kingdoms once... and presumably each king minted his own coins. So expect to see references to halfpennies, threepennies, stars, and groats popping up in future books.
-http://www.westeros.org/Citadel/SSM/Entry/1243/.