The best advice can only come from someone who knows what your current soil is like and what it lacks. But here are some differences between the amendments.

Mushroom manure adds organic material without a lot of nutrients, so you might say it is fairly "nutrient neutral". It would be good in most soils, particularly if they are either sandy or full of clay and in need of additional organic material. You would then add other amendments or fertilizer to balance any nutrient deficiencies in the soil. The risks are low and it is helpful for many soil types.

Lifestock manure when well composted also adds organic matter but is also high in nitrogen. So it is like adding compost + a high nitrogen fertilizer. It is a good choice if you need the nitrogen. But the manure has to be well composted before you add it to the soil. If you add horse or cow manure directly to the soil the ammonia and salts that are released as it begins to decompose can be too much for your plants and burn them. It can take 9 months to a year before the manure has composted down to the point where it is safe. So find out how it was produced.

Chicken manure is livestock manure on steroids. It is higher in nitrogen, which can be a plus if that is a deficiency you need to address. Again, it has to be composted (or "aged") before it is safe to add to an active garden.

In short, if you don't know your soil shortcomings, mushroom manure is always the safe bet - especially if you are going to use a specific rose fertilizer to supply nutrients. If you are going to fertilize anyway, then the potent livestock or chicken manure may be overkill.

Whatever you buy, if you're planting in it immediately, make sure it's fully composted. Not steamy or smelly.

When buying products I often grab several types to get a good balance between nutrients, texture and cost.

Thank you so much for your input toxcrusadr and Richard! I am going to go for mushroom compost with some livestock manure but will check that it is fully composted!