You can get carer’s allowance without being on UC, but if you are working full-time your earnings will be over the limit.
You could relook at UC. Some people with a disabled child are eligible for UC even if they both work.
If DC is in receipt of high rate mobility, DC is eligible for a blue badge. Even if DC isn’t in receipt of HRM, you can still apply and be assessed.
If DC is in receipt of HRM, depending on where you live, there may be a taxicard scheme DC is eligible for. For an example of what I mean, have a look at the London taxicard.
You could see if DC is eligible for your local travel concessionary pass. Eligibility will depend on where you are and what DLA rates have been awarded.
Lots of places offer a free carer/companion ticket. Eligibility criteria and proof accepted varies place to place. Some accept DLA, some don’t. Not everyone chooses to get an access card. Even for places (e.g. some theatres for companion tickets and Merlin for a RAP) who require you to go via Nimbus don’t require you to pay for an access card.
Some places offer reduced price tickets for those on benefits.
There are various grants for DC with disabilities. Not all linked to DLA itself. If there is anything specific you are needing, post and someone may have suggestions. Some not in receipt of UC are still eligible for Family Fund.