Here is the result of my own research into the question. In summary, I failed to find any materials online to substantiate the claim that Mikhail Bakunin was a murderer or that he bombed anarchists, or bombed anyone at all. However, I also found a conspicuous absence of information on what he did during his involvement in various uprisings in Europe in the 1840s and afterwards. If not for the political and philosophical implications, I might conclude the claim is false presumptively; but there is an apparent conflict of interest at play that gives me pause.
I would conclude the claim is false if I were to find any of these, but I have found none so far:
- A sympathetic or neutral source on Bakunin that documents fine-grained details of his participation in the uprisings and "abortive revolutions" of the 1840s, that does not include any murders or bombings
- A critical source on Bakunin that criticizes others for their participation in leftist violent uprisings, but criticizes Bakunin only for his ideas
- Documentation of past, publicized claims against Bakunin, contemporary to his time or shortly afterwards, in which he was accused of murder or bombings, that were later debunked
The use of "some believe" in the claim is clearly suspicious, though perhaps forgivable if the claim is in fact true; after all it is just a card game. But I hesitate to dismiss the claim out of hand for two main reasons.
First, the rest of the card's bio seems to check out, as documented in the question. Even the first part of the questionable claim (that he "called for the abolition of all state structures") seems to be justified by discussions I found of his philosophical differences with Karl Marx (see "Marx" section) and his other contemporaries.
Secondly, and crucially, I found little discussion of what Bakunin did other than write and talk about these issues during his life. And yet, according to this source:
Bakunin ... saw himself primarily as a man of action, although his action was rarely successful and his life was punctuated by abortive revolutions.
The most specific information I can find on his actions is from the same source:
The years of the revolutions in Europe -- 1848-1849 -- were the most dramatic period of Bakunin's life. He was an enthusiastic partisan of the uprising in France; later in 1848 he fought on the barricades of Prague, and in March 1849, he took a leading part, with Richard Wagner, in the Dresden revolution. He was captured there and, after periods in Saxon and Austrian prisons and twice being sentenced to death and reprieved, he was handed over to the Russian authorities, who imprisoned him in the Peter and Paul Fortress. Six years there ruined his health. In 1857 he was sent to exile in Siberia, and in 1861 he escaped, via Japan and the United States, to western Europe.
So he "was an enthusiastic partisan" in one uprising, "fought" in another and "took a leading part" in another. This leaves me to wonder, what does that mean exactly? It's noteworthy enough to draw multiple death sentences and hard labor in exile, and mention in multiple modern biographical articles, and yet completely non-specific in the online record as far as I can discover. It's weird.
In comparison, many specific details of Bakunin's social contacts are documented from the same period. Including this fawning commentary from Wikipedia on the final stages of his impressive escape from Siberia to western Europe:
In Boston, Bakunin visited Karol Forster, a partisan of Ludwik
Mieroslawski during the 1848 Revolution in Paris, and caught up with
other "Forty-Eighters", veterans of the 1848 revolutions in Europe,
such as Friedrich Kapp. He then sailed for Liverpool arriving on
December 27. Bakunin immediately went to London to see Herzen. That
evening he burst into the drawing-room where the family was having
supper. "What! Are you sitting down eating oysters! Well! Tell me the
news. What is happening, and where?!"
To see what I was hoping to find, many specific details of Fidel Castro's invasion of Cuba are documented, including certain of Che Guevara's specific actions in the heat of battle. Had I found similar fine-grained details of Bakunin's exploits, I could more easily conclude that the claim by Steve Jackson Games is false.
But with this gap, the claim amounts to an unsubstantiated rumor on which the historical/biographical data I can find is fascinatingly silent. Despite the description of Bakunin as a "man of action" who believes that destruction is a force of creation, and despite many allusions to his involvement in multiple insurrections, there seems to be a glaring omission in online materials of what exactly his actions were.
I should say that all of the sources I have considered for biographical and literary study of Bakunin are either neutral or sympathetic: Wikipedia, academic-looking sites, and (apparently) socialism or anarchism sites. I have not found any information on Bakunin that takes a critical or contrary viewpoint of his writings or deeds. Even finding a website highly critical of Bakunin's ideas, and that criticizes the deeds of Che Guevara and other leftists, but does not mention Bakunin being a murderer or bomber, would probably convince me the claim is false. But I have found nothing like this either.
Could it be that those who might devote time and resources to documenting the written works and biography of Bakunin online might also choose to omit certain acts on his part that might today be viewed disapprovingly? Is there only oral tradition, and no documentary source, remaining of Bakunin's acts which somehow found its way onto this card? Or did Steve Jackson Games simply invent this murderer/bomber description of Bakunin out of whole cloth?
Right now, I'm not really sure which to be more skeptical of: the game card, or the treatment of Bakunin in online sources in a strictly respectful and academic tone, lacking fine-grained details of his actions during the various civil unrests he participated in.