Several things you could do come to mind.
I have to carefully mound it up before tamping and inevitably some small amount spills before I have tamped.
Check how much coffee the filter inside your portafilter is rated for. The Bambino Plus uses a non standard 54mm portafilter and from what I can see, you should have received 4 different filters. Only use the ones with straight walls and if your grounds are flowing over, you are probably using too much coffee. They say it can handle up to 18g in the largest basket but that feels like it would be too much.
Slight excess after tamping. Despite weighing the exact dose I usually end up with a bit extra.
What do you mean you end up with a bit extra? When you tamp the grounds, it should be compressed and at that point the only thing you should do is brew the coffee and not mess around with the already prepped puck.
often the portafilter basket itself comes out
You're supposed to knock with the basket towards the rubber, knocking the grounds loose and then just lift it to allow them to fall out. That way there is no possibility of the basket flying out. You will never be able to get every single piece of coffee out, which is why most people use a microfiber cloth to clean out the basket.
there’s always some grounds in the portafilter along with some water
Next time after knocking out the grounds and wiping the basket with a cloth, remove the basket and check if there are grounds in the space between the basket and the spout. If there are a lot of grounds there, your filter basket might have too many large holes or you are grinding too finely. If there is nothing there, check your brew head. It might be full of grounds, which is mostly caused by using too much coffee in a shot. You can clean that with a simple brush. Make sure there is no coffee stuck to the brew head as it might get rancid over time and negatively impact the taste of your coffee.
You might want to check out this video series on espresso by James Hoffmann, where he goes into every aspect of brewing espresso. Seeing what he does differently might give you an indication as to which things you can change or adjust to make your experience more pleasant.
Edit to answer the comment:
the tamp doesn't fully occupy the width of the basket
You should really consider getting a tamper that is suitable for your basket, there should be barely any room between the tamper and the walls - ideally, just enough to prevent an airtight seal. It really helps to ensure even flow and it prevents loose grounds floating on top of your puck when you remove it after a brew.
if I knock immediately after a shot it still makes some mess because there's a small amount of liquid in the basket
A little bit of liquid is normal but this might also be caused by the loose coffee that wasn't tamped. It could also be caused by pulling a shot that is too long. In general, try to aim for a ratio between 1:2 (18g in 26g out) and 1:3 (18g in 44g out).
