If we cover the surface of the moon with a four kilometre deep sea of blood - at the traditional temperature of 37°C - it will appear to be red... for a little while:

Image from http://blog.toddbodnar.com/2010/05/ray-tracing-for-beginners.html
The problem is that blood is not an inert solid, it's a suspension of red blood cells in water. Now, let's consider that the surface of the moon ranges in temperature between -133°C at night and 121°C during the day. It's also in the vacuum of space, and the moon's gravity is not strong enough to hold on to all that water.
Pretty much immediately, the water is going to start to boil and sublime away into space, while the blood cells are heavy, and will start to settle to the bottom of the blood sea. Since the moon has some gravity, the moon will be surrounded by a thin atmosphere of water vapour that will act to slightly increase the pressure on all that liquid water while it freezes.
On the dark side of the moon, where it's cold, the surface of the lunar blood sea will begin to freeze at the same time that it boils, forming a white rime of frost and ice... maybe with some blood cells in it, but not many. The ice may appear to be a pale pink. However, since the dark side of the moon is... well... dark, we won't really see that... we'll just see it reflecting the light from Earth, so it'll appear a slightly brighter dark grey than usual, but still dark.
Meanwhile, on the light side of the moon, the surface of the blood sea will be sublimating more rapidly into the vacuum of space... but still beginning to freeze. While the surface of the moon is above the boiling point of water at 1 ATM, the amount of energy reaching a given area of the surface of the moon is about the same as that reaching Earth, so the entire depth of the blood sea isn't going to boil like a pot on a stove. However, with the blood cells settling out of suspension, the surface will boil until its temperature drops low enough for it to freeze.
At that point, we'll have a moon covered in a layer of ice. With the hottest water molecules continually sublimating off into space, the temperature is going to continue to drop, and the blood is going to freeze from the top down.
When the moon is completely covered in ice, it'll no longer look red, but white... or pinkish white. Blood boiling and freeze-drying on the moon will not behave like blood freezing at low temperatures on Earth, where it can remain red.
The blood ice covering the moon will take a long time to sublimate away, but given that the moon is in the liquid water zone of the sun, it will happen eventually. However, the ice will have increased the albedo of the moon, so less of the sun's incident energy will be retained at first. This would be like an ice-age on Earth, where ice prevents its own melting.
However, over the course of years or centuries, the ice will sublimate away until the settled red blood cells at the bottom of the frozen blood sea are revealed. At this point, the colour of the moon will begin to change again... to that of freeze-dried blood:

Image from https://www.danubeadornments.com/product-page/freeze-dried-blood
Instead of grey lunar regolith, we'll have reddish-brown freeze-dried blood. At first, it'll be mottled reddish-brown and white, since the ice won't sublimate away evenly, but eventually it'll become entirely reddish brown. That'll probably take well over a human lifetime to occur.
So, to sum up, yes, the moonlight will be red for a little while, until the blood freezes over, which should happen within a day. Then it will be back to white (or pinkish white) for many years or even centuries, before becoming reddish brown, or at least redder than it was before being covered with blood, but not as red as it was in the first moment after the blood was deposited.
Edit in response to the question edit:
As to whether there would be visibly red light from the moon on Earth... that's likely to be subjective.
A blood-drenched moon will have a higher reflectivity than the pre-blood moon, and it is possible that the light levels will be bright enough that many people would see red light from the moon at night before all that blood freezes.
Then, after the blood freezes, forming a pink-white crust on the moon, you'll just have particularly bright moonlight... it might be a little pinker than normal, but it may not be perceptible to many people.
Once the ice sublimates, leaving a surface of freeze-dried blood, the moon will be less reflective than it was when it was icy, and it may not be bright enough for any reddish tint to be visible to humans on Earth.