Animal Welfare Professionals

We are a small foster-based rescue, but quickly growing. Budget has always been a hard topic with our leadership, so I'm hoping you can offer some insight on two questions.

We have our budget income broken down into line items for individual fundraisers/campaigns, donations for boarding (because we have a few people who donate specifically for that), individual grants, adoption fees for each species, and then a category called "donations from individuals" which is kind of a bucket for everything else. We have about half a year of expenses in savings because I'm super cautious.

Question 1:

We take care of dogs and cats. On the dog side, they feel like money donated from a "dog person" should go in a category that support dogs. They think the expectation from the giver is that they are supporting dogs, not cats or the the organization as a whole. They have even asked the givers to specify support of dogs so it's categorized that way. They see that as "their" money for dog causes.

When the cat people get extra funds (like someone gives some money when we take in a kitty, etc), it's categorized as "donations from individuals".  That money goes to support the overall budget because our adoption fees do not cover the true cost of care and running of the organization.  It's not designated only for cat causes, and the person receiving the money doesn't push the giver to designate as that.

This has been a big source of stress for me since I do the books and get the brunt of the hate. I feel like we are one organization and people who give (except for a specific fundraising campaign or grant), are simply supporting our overall mission. I would like to know how other orgs deal with things like this.

Question 2 -
We set our expected budget at the beginning of the year with income goals and expenditure estimates. For example, we have a fundraiser for a certain cause that we hope will make $5,000. The overall expected budget expenditure for that cause is $15,000, hoping to make $5,000 at the fundraiser and the rest would come from other donations or savings.
If we raise $6,000 fundraising, does the budget automatically go up to $16,000? Or does the extra $1,000 help fill that $10,000 hole?

How we have it now, the group overseeing that cause could go back to the board and say "hey, we raised more than we thought. may we up the budget?". The board would look at our overall donations and expenditures, see if we're on track, and then approve if everything is good. If we are falling behind expectations, budget would remaine the same.
But that group thinks the budget should automatically increase. They said if it doesn't increase, then why should they bother to fundraise - they'll get the same amount of money either way ($15,000). 

I hope this makes sense. I'm open to any ideas that might help lower my blood pressure! Would love to hear how other smaller, multi-species rescues do it.