I can do copy and cut operations from icloud.com apps, so why not paste? Maybe that works for you, but you're tired of not being able to paste into certain login forms around the web. The solution in either case is simple:
Don't paste your clipboard contents ⸺ replay it.
First, you'll need AutoHotkey installed. I could try to explain it, but Wikipedia does it better:
AutoHotkey is a free and open-source custom scripting language for Microsoft Windows, primarily designed to provide easy keyboard shortcuts or hotkeys, fast macro-creation and software automation to allow users of most computer skill levels to automate repetitive tasks in any Windows application.
Once it's installed, right-click somewhere in a folder and select New > AutoHotkey Script. Open it in your favorite text editor. It will contain a few helpful setup lines that you can leave in place and ignore. Below them, add this:
!v::SendInput, %Clipboard%
Yes, a single line. Save the file and double-click it to launch it. This little script will live in your tray until you exit it. That's all there is to it! Now, whenever you're faced with a place you can't paste text for one reason or another, hit Alt-V instead of Ctrl-V.
In telling AutoHotkey to "send input", it will literally send the text on your clipboard, referenced by %Clipboard%, as though you were typing those letters yourself. Since you're not doing a "paste" action anymore, all restrictions blocking it are no longer relevant.
Bonus tip time!
Add a quick second line to your little script to turn your otherwise useless Caps Lock key into an alias of your left mouse button. Mouse-clicking index finger getting tired? Use Caps Lock instead! This doesn't stop your left mouse button from working; it simply adds another way to do what it does.
CapsLock::LButton
There's a hell of a lot more you can do with AutoHotkey, such as changing the script's tray icon into something other than its goofy Comic Sans logo, so I encourage you to explore its documentation. It's very neat, though the syntax kinda sucks, in my opinion.