Good stuff Paul.
Sometimes when I struggle with finding a rhyme I change the word I'm trying to ryhme with. Try that but keep the same sentiment and it may open up the possibilities. Right now your looking for a ryme with heart. So you have .....
art, artcc, arĂȘte, bart, cart, carte, chart, charte, clart, dart, dartt, fart, haart, hardt, hart, harte, hartt, mahrt, mart, marte, part, parte, scart, schardt, skart, smart, smartt, start, tart, tarte, tartt, apart.
Nothing great there, you can get by with "make a fresh start" or "I'll play my part" but we don't just want to get by, we want to be awesome.
But you know I'm always here
To catch you when you fall (or)
And I'll watch over you (or)
To mend just what you break (or)
To hold your heart with mine (or)
To Hold your heart in my hands (or)
To open every door (this one is working backwards from the last line to get a rhyme)
To mend your broken heart
Just leave it with me honey
Ain't that what friends are for
Hopefully you get where I'm going with this. Sometimes some words are not strong rhyming words and don't give you what you need to go forward with the song. You'll find most times people write a song they will rhyme heart with apart. As in I gave you my heart and you pulled it apart. Avoid the obvious.
Good luck.