Unfortunately, after days of searching for an answer to this, I cannot find anything that explicitly states that pasting text is supported, or unsupported in a native way. However, the general consensus everywhere I land seems to be simply an inference that it's not possible.
With that, I wanted to create an application that would simulate typing so that you could replicate pasting the text. However, per the Rocket League terms of service, this would fall under "Prohibited User Conduct":
(p) develops or allows any viruses, malware, adware, spyware, Trojan Horses, cancelbots, other automated “bots”, “auto players”, cheat utilities, spoofers, keyloggers, or other circumvention devices to be present on any machine used to access the Services; and/or to attempts to upload/utilize such devices against the Services;
As a result, I'm forced to give up my pursuit, and concur with the inferred consensus that it is simply not possible.