"Any more traditional anti-monarchist songs?"
Can't lay my hands on it at present, but I have a recording somewhere of MacColl singing the broadside "The Brass Warming Pan", which suggests that the powers-that-be of the time produced a male heir to the throne by slipping one into the queen's bed in the aforementioned vessel.
Not necessarily traditional, and probably not singable, but some of the earliest songs on royalty are to be found in Thomas Wright's 'The Political Songs of England - from the reign of John to that of Edward II'. Most are in either Latin or early French, but they come with summaries in English, and all are well annotated.
Also worth looking out is 'A Book of Scotish Pasquils (satirical pieces) 1868 (no named author, but it has been attributed to Maidment).
Peggy Seeger used to sing a beautiful ballad called "Fair Rosamund" about the seduction of Henry II's mistress Rosamund, who was eventually poisoned Queen Eleanor.
From memory:

I have a sister, young Clifford, he says,
A sister no-one knows..
She has a colour in her cheeks,
Like drops of blood in snow,
Like drops of blood in snow.

She has a waist, a waist, a waist,
Like to my silver cane,
And I wouldn't for ten thousand pounds
King Henry know her name
King Henry know her name.

The king being up in a bower so high,
Hidden close and still,
And every word young Clifford spoke,
He wrote down in a bill
He wrote down in a bill.

..... cant remember verse

The first fair line that she looked on,
It made her laugh and smile
And the second line that she looked on
The tears they did run down,
The tears they did run down.
Cursed be my brother Clifford,
Cursed may he be,
Why can't he dote on horse or hounds,
That he must dote on me?
That he must dote on me?

My own personal favourite about the gettings-up-to of royalty.


QUEEN ELEANOR'S CONFESSION.

Queen Eleanor's sick and she's very, very sick,
She's sick and like to die,
And she has sent for two friars from France
To pardon her sins e're she die.

When the king came to hear of this
And angry man was he.
And he has sent for the Earl Marshall
And bid him come speedily.

Now you'll put on a friar's robe
And I'll put on another.
And we'll go in before the queen
To pardon her sins together.

Oh no, oh no, cried the Earl Marshall
Such thing can never be,
For if the queen got word of this
High hanged I will be.

Now I swear by the hilt of my good broadsword
And by the heavens so high,
That not one drop of your blood shall be spilt,
Earl Marshall you shall not die.

So he's put on a friar's robe
And the king's put on another,
And they've gone in before the queen
To pardon her sins together.

Now the first great sin that ever I did
I will to thee unfold
Earl Marshall had my maidenhead
And the truth to you I have told.

Now that was a sin and a very great sin,
But pardoned it must be
Amen, amen, cried Earl Marshall,
And a very feared man was he.

Now the second great sin that ever I did,
And a grave sin it was too;
For we've carried poison for seven years
To poison King Henry.

Now that was a sin and a very great sin,
But pardoned it must be.
Amen, amen, cried the Earl Marshall
And a very feared man was he.

Now had I not sworn by the hilt of my sword,
And by the heavens so high,
That not one drop of your blood would be spilt,
Earl Marshall you would hang high.

Now she's turned her head away from the king,
With her face turned to the wall.
And they've heard no more of her secret sins,
That she had not been a maiden at all.