Well, I think both [name]Cate[/name] and [name]Kate[/name] have plus points. I think [name]Cate[/name] looks softer, more elegant, and is more unusual (I always meet Kates, never Cates), however if you plan to use the full name [name]Catherine[/name]/[name]Katherine[/name], the K spelling seems more interesting and spunky to me (also, I love the nn [name]Kitty[/name], which doesn’t really work as Citty).
I do like [name]Cate[/name]. I actually prefer it to the full [name]Catherine[/name], even though I would normally caution against using a nickname as a full given name. I don’t think there are any spellings other than [name]Cate[/name] and [name]Kate[/name] which have the same refined and classic feel.
If you’re going for just [name]Cate[/name], I think a long middle name is in order, preferably three or more syllables. Something unusual would compliment the classic [name]Cate[/name] well, too, and vowel names are the easiest flow-wise.
Some examples :
If I’m honest, I don’t think [name]Cate[/name] [name]Miley[/name] flows well, and I don’t really like [name]Miley[/name] at all. It has no history or meaning behind it, and I think names are somewhat empty if all they are is a string of nice but meaningless sounds.
Personally, I like middles that have special meaning. That could be using the name of someone important to you, but there’s no point using the name of someone you love if you really dislike it - you can still honour them through a different name, maybe one which shares the meaning as their name, or is related sound-wise (e.g. [name]Louise[/name] becomes [name]Lucy[/name], [name]Susannah[/name] becomes [name]Rosanna[/name]). Letting your parents choose a middle would, I’m sure, be another great option, as I imagine they would be flattered that you trust their judgement so much and feel a special connection with their granddaughter because they helped to name her.
Special meaning could also be a name from another source, though - the name of a constellation because you love astronomy, a name from your favourite song, the name of the place you and your partner honeymooned, a name from your heritage, a spice name because you’re a chef, anything like that.