Is offering to use personal funds for research a good idea?
I think you shouldn't offer private fund to supplement a research budget, and I also think people won't let you for the two following reasons (among others):

1. It's not your private responsibility to fund your research. For the sake of your own self respect, you shouldn't pay to work. It will give a bad signal, even subjectively, about the quality of your work. It's important for an academic career to be able to show that you can secure complete funding for your research.

2. Personal funding is not reliable:
People reviewing your application will see that the project is underfunded and will doubt that you will effectively be able to compensate with your own money. There will be no ways for the institution to enforce a funding commitment like "the project will be supplemented by personal funds". Funding is generally secured via contracts or other forms of legal documents.
Besides, any institution that relies on researchers breaking their piggy banks to pay for equipment is doing a sloppy job and it's going to hurt its reputation.