NannyRodder I am going to be blunt here - you don't know me, so may be it easier for me to say it.
You sound as if you think that time for you with your GD is a right - it is not, it is a privilege, a gift that it is not for you to demand.
The idea that you lend them money and that this might encourage them to give you time with the babe as a thank you is not a good mindset.
If you will forgive me saying so, this is all about YOU - your idea of helping them is to satisfy your needs and not theirs.
Grandmothers are in a funny situation - we love our GC as if we had borne them ourselves and we have to grit our teeth and stay back and involve ourselves on someone else's terms - it is very hard and is the first and most difficult lesson of grandparenthood.
I may sound hard-hearted, but believe me I do feel for you and do not want to see this situation continue.
You have to play it by their rules - you have no rights. Make sure that every approach you make is truly for their benefit. Be cheerful and supportive, take what comes your way with a smile; and take the knocks with a smile too - or you will alienate them and the situation will get worse, which is the last thing you want.
You cannot have a "lend" of your DGD - she is not a commodity to be traded with financial loans. Their request that you come over and get to know her well is a generous and reasonable one. Grab it with both hands and go and see her with love, not resentment, in your heart.
The MIL/DIL relationship is a tricky one and you must play it carefully if you are going to get the chance to enjoy your DGD and she is to get the chance to know you and experience all you can offer.
You have already had her overnight once, and instead of reacting to this with joy, you see yourself as a last resort - so starting with that resentful frame of mind will not help.
None of my DGC stayed with me till they were 18 months or more - so you are a very lucky nanny!!