Is there any way to get out of the ruins with the name Frisk?

Apparently, no.

When you name yourself Frisk, the first thing that appears is a note from the developer, Toby Fox. The message says:

“WARNING: This name will make your life hell. Proceed anyway?”

The name Frisk enables Hard Mode, and all the Ruins monsters are replaced by stronger counterparts.

When you defeat Toriel (whether it be by kill or spare), the Annoying Dog will appear and say, "And that’s the end of Hard Mode!"

If you tried betrayal killing Toriel, she will stand up and go back to normal. This will provoke Annoying Dog into saying, "Shouldn’t you be dying or something?"
To this, Toriel responds that it doesn’t matter anymore, because you reached the end of Hard Mode. After this, the screen will do a mocking of the Undertale opening by instead showing, "Undertale: HARD MODE…Coming Soon?…………………….eh…don’t count on it."

Flowey will eventually appear if you stay looking at the screen and ask you things like, "Do you have anything better to do?" and "Nothing else is going to happen here. Please exit already."

Since Hard Mode ends at the Ruins, you are required to rename yourself when you reenter so that you can proceed.

_{Source, Quora.}

No, Whenever you get to the end of the ruins you have the option to either kill or spare Toriel, if you spare her the annoying dog will appear and say that's the end of hard mode if you kill her she gets up anyway the annoying dog appears and says shouldn't you be dead or something the conversation continues and the normal outro begins. you have to rename yourself in order to get to hotland.