she is idealising something that isnt real though - if i am reading correctly, then from what you have said, the contact has been very minimal - not someone around all the time at all and really only when you facilitated it.
curious as to why she has such a fantasy about her daddy?
i dont think you should compromise her safety tho by trying to get more visitation -after all if he is a) unwell and b) has poor track record in making effort then you will only be setting her up for even more disappointment.
maybe some serious talking about how he is unwell - has she seen/witnessed anything at all you could use as a reference point to see if she understands how unwell he can be?
in any case - he should be making the effort - you ahve valid reasons to not allow contact, let hi take it further eg to court adn apply for visitng rights. if he does not do so - why are you setting it all up? it will then fail again at some point.
my youngest was five when my exP had his violent psychiatric episode - for a long time she said she didnt want to see him, was scared of him (he waent aborad his home country). then she idealised him, wanted him back, was so happy when he returned (he manipulated into staying in family home) - then he became abusive towards me and more scary times so she was glad when we moved away. she has said many times we should all live together, talks about mummy and daddy kissing an getting married etc.
while the older dd is pretty clear about not seeing him etc.
so - in roundabout way- i think six year old girls do idolise/idealise their daddy. she loves him - but what does she love? he sint there day to day is he? he has been erratic and sparse in visits... she is chasing a fantasy daddy....
of course she has rights to see him but it has to be safe, dont be drawn into thinking if you let her see him more it will solve her upsets, it might be a ) not safe and b) might in long term cause more upset given his record... you know that you are not stoppping contact, he is just unwilling to make the extra effort or abide by rules - my ex chose to not see his dcs for a few months because he refused to accept sharing costs with me for contact centre - so he worked to get CAFCASS to pay - which they did but had he raised the money at once he would not ahve had to wait for three months to see them - none of it makes sense. it was about pwoer games with me....
i would actually see if there was access to child psychologists in your local area - ask GP. a good family therapist who works with children eg play therapist might eb abel to talk and draw out what it reality about what she wants and realistically achieveable adn what is fantasy..and maybe help you come p with strategies.
in my case it ahs been fairly easy to jsut keep aying "it isnt possible for mumy and daddy to live together" - and i have back up if you like from her big sister who understands it all so much more...but i can see the six, seven year old who fantasises about a happy mummy-daddy family....
it is hard. but you need to be realsitic and put safety and reality above what seems to be an easy solution ie set up and enable the contact so you are doing it it, or take risks...
we cant protect them from how their otehr parent actually is, we need to try and help them separate reality from fantasy...and that might need professional inppuit from someone like a family therapist
i totally understand the heartbreak at seeing her cry over him " 'the only thing that would help is to see daddy again'" - but maybe getting to the bottom of how/why she is thinking this might help? you know that "seeing daddy again" is not going to solve her worries/concerns - he is always going to be someone who is not - by his track record - going to be consistently in her life and that is something she/you have to accept. ...
if he has another major episode and is secitoned - how will you support her then? is she believes that he is something he is not - it is very difficult for sure. but it is not as simple as "she loves her daddy and should see him" - when dady is someone who lets people down (his track record) and has untreated(?) mental health isues.
is also hard to explain "ill" when it is mental health/more than "he has flu" i think.
see if you can get GP referral to family counsellor/family therapist/child psychologist for advice and strtegies.
in meantime let him seek court order/get cafcass involved etc..