I was all set to say “Go” - having lost both my parents and being at a distance, I am so relieved that in both cases I was able to do this. And, unless your mother is still physically able to enjoy her grandchildren, I would have said to leave your children with their dad because you need to be with your mum, not entertaining a four year old.
Reading on though, a couple of things have changed my opinion somewhat.
The main issue is that your DD would be staying with someone who bullies her?? That’s not what you said in the OP, and makes a massive difference. What you described happening is just not on. And over 6 weeks you don’t know how that would develop. You need to find a way to protect her from that, preferably by not leaving her there.
Also, the “living on take-aways” is unimportant in one way, but when your reason for not taking the children is money, that seems odd, as someone else pointed out: takeaways are not a cheap option. Or do you mean they’d live on cereal and crisps?😃
More seriously - couldn’t that budget be put towards taking one or both with you? And could the 6 year old go to a local Spanish school? She might not learn as much as in her own school but she’d be in school and would be a great experience for her? She’s only six - she can catch up on her English speaking education afterwards.
ETA: I forgot the easiest one to fix - FaceTime (or WhatsApp video chat etc). We’ve done this many times for older family members, including us adult cousins organising Christmas and birthday “video meets” for the older generation (my mum and her siblings) during Covid and when they could no longer travel easily. IOW agent your sister or someone else to set up a WhatsApp chat for your mum and you. Easiest thing in the world.