I did a little digging on the internet...since your T doesn't appear to be bleeding out and looks overall very healthy (still climbing and has strength), I would make sure it has clean water to replace any lost hemolymph and leave it be. Below are some of the different sources I consulted - their reputability is really opinion-based so take whatever is gleaned with a grain of salt. Archnoboards helped too. The consensus is have corn starch and super/krazy glue on hand for bleeding cuts...
http://arachnophiliac.info/burrow/1staid.htm
The problem next is wounds. Spiders have no coagulant in their blood, and this means even a slight wound will not scab over and can lead to death. Again, prevention is better than cure, so keep sharp objects out of the spiders way, and low-sided tanks are better than high for terrestrial species, because your spider will climb and a fall could cause an injury - usually a broken leg or spinneret. Bleeding wounds should be treated with a non-scented talcum powder applied directly to the cut, and then left alone. Do not try to remove the scab as you will only re-open the wound. Damaged legs or palps should not be treated at all, as the spider will shed these and grow new ones if it needs to. If the wound bled badly, make sure the spider has plenty to drink, and as soon as the bleeding has stopped and the scab has set firm then offer food. This will replace fluids quickly. If the wound is too large to treat and it is obvious the spider is going to die then place it in the fridge for 15 minutes, and then into the freezer. This will end its suffering quickly.
https://regalpet.com/pet-health/876-tarantulas-injuries
Tarantulas can also suffer from cracks or breaks in their body. Body cracks and breaks can be sealed with spray dressing or petroleum jelly. If the cracks or breaks are minor, they should be treated with spray dressing or petroleum jelly, and they will usually heal themselves when the tarantula molts. However, severe cracks and breaks in the body are usually untreatable and molting will not help heal them. Unfortunately, tarantulas with severe cracks or breaks on their body often need to be humanely euthanized.
http://www.tarantulas.com/first_aid.html
SUPER GLUE AND INJURY REPAIR
Household super glue is the most essential item in the tarantula keeper's first aid kit. If your tarantula is injured due to a fall or other mishap, having super glue to mend the wound and clot the flow of hemolymph is imperative. Hemolymph is the tarantula's "blood" and it is a clearish-whitish-pale bluish liquid (depending on who is describing it). Careful application of super glue can even repair ruptures to the abdomen if the injury is caught immediately.
When spiders lose legs, they typically lose them at predetermined ‘break points’…usually at a joint close to the body. These points have special muscles which clamp shut, to ensure the spider doesn’t lose too much blood. This process is called ‘autotomy’, and it’s completely normal...
... However…what happens if the leg is amputated before the break-point?
Well, the spider ends up losing a lot more blood before they’re able to eject the leg. When a spider loses too much blood, it has trouble moving. Often, injuries can remove so much blood that the spider goes into a stereotypical pose called a ‘death curl’ when things get really bad. Since extension of the leg is caused by an increase in blood pressure to the limb, the death curl is what happens when the spider’s blood pressure gets too low for it to extend the limbs...