Lilypond has few restrictions, but one is that it does not currently support slurs across voices. Refer to the Notation Reference manual Snippets sections for the topic 'Creating slurs across voices'. It shows how to use invisible notes to do this.
I am compelled to say that lilypond can be made to do almost anything imaginable in scores, but it is not a tool for a person in a hurry. That's not the nature of it. It's a precision tool of great subtlety and power. I engrave very complex scores for a colleague of the New Complexity School (although he would deny this appellation). Perhaps my scores are similar to yours. They are quite some distance from the Common Era tradition which lilypond is mostly based on. I found it is really worth the investment of time and effort to learn Scheme and how to extend lilypond. This then gives you a great deal of freedom to make quite complex constructs. And if you need complex graphical work, you can always use Postscript in lilypond as well.
I'd recommend doing as much in lilypond as possible, without going outside to Inkscape, because then you can make edits and changes easily - a difficult workflow if using an external graphical editor. I used to use Inkscape post-engraving, but abandoned it when I realised you just have to sit down and learn Scheme in lilypond.
I am aware that you may be time poor, but if you are committed to lilypond it's worth spending the time, step by step, learning Scheme. It will repay you over and over. The best thing is that there are many very learned colleagues here on the list who are always willing to answer technical questions. There are no stupid questions! Once you build a library of code for your special work, then you will be able to go back to being in a hurry!