The way I see it the Putin investigations is an attempt to "pin something nefarious" on The West, hence the so called issue with British intelligence. Ukraine is a member of the European Union and has been building that relationship for a long time. The focus of the relationship was economic. Ukraine and it's business community wanted to open economic relationships with the European Union.
The point is regardless of how legitimate the relationship with the EU was and is, Putin will make it look oppressive and in hate of Russians. These investigations are not for the world to understand, but, a closed audience in Russia.
What should really occur is for Putin, his lawyers and his foreign ministers to take their case to The Hague and argue against the charges that were filed. He should do that in person to demonstrate he is not afraid of the truth.
No investigation by Russia will result in anything except a narrow focus for propaganda reflecting his politics.
He does want to escalate the war. If The West gives him no reason to act, then he will create a scenario in order to justify it to the Russian people.