I've spent the morning trying to find any contemporary mentions of Oliver Law, and the most likely answer is that outside of very specific, narrow circles of socialist and communist activists, and segments of the African-American community, knowledge of Oliver Law commanding the Abe Lincolns would have been entirely lacking. He only took over command in late June, and was dead by mid-July and as such the only mentions I could find in any major newspapers during his lifetime were brief notices of his death, likely the first point anyone had heard the name. This was the wire report to the New York Times (which was republished in several other papers):

Madrid, July 13. - Oliver Law, Negro commander of the Abraham Lincoln Battalion, was among a small number of Americans killed storming Villanueva del Pardillo last Tuesday, it was learned today.

American officers have had hard luck during the great offensive. Martin Hourihan was badly wounded. Then Law was killed, and the two lieutenants who took over successfully in the field were both wounded.

However, the battalion as a whole did not suffer heavy losses in this operation. The only other death reported in the last few days was that of Max Krautheimer, New York lawyer. Jean Bronstein's death had been previously reported.

Similarly the New York Herald Tribune ran a brief blurb from the UP wire, but they didn't even call him the commander, simply noting he was an officer:

Madrid, July 12 (UP) - Oliver Law, a Negro office of the American volunteers in the International Brigade, was killed in the fighting around Villanueva del Pardillo, the War Office announced tonight. Law, twice cited for bravery, was an expert machine gunner, trained in the United States. He was the last survivor of eight Negros who arrived in Barcelona last November to form the first American volunteer group with the loyalists.

Some more involved mentions can be found in Black newspapers, such as the Baltimore Afro-American, Norfolk New Journal and Guide or The Chicago Defender, the latter proudly noting that Oliver was "in full command [...] when he fell mortally wounded", but even here all mentions I can find came after his death and use him for rhetorical effect, and they would be more limited in reach than a paper such as the Times. The only publications which spoke of Law prior to his death were CPUSA affiliated, principally The Daily Worker, which ran a piece about his performance at Jarama in April of 1937 as part of their efforts to improve African-American outreach, but this would have been quite alien to most Americans of any race.

Now, to be sure, there were racial tensions here, but they were quite narrow and limited. Law had claimed six years experience in the US Army, as a sergeant, which helped him quickly rise in the ranks of the Abe Lincolns, but not everyone believed his claims (and rightly so, as he never rose above PFC), and seeing him pass over others who had been there longer caused some bad blood. Some saw him as being fast-tracked due to his race (as opposed to his claims of experience, but the two are hard to separate out), and the afore mentioned efforts by the CPUSA to appeal to African-Americans. At least some soldiers took on as a slogan "Restore the whites to equality with the Negro", but plenty more respected him well enough as a soldier, even if not as a competent commander. In the end, there are many who spoke poorly of his skills there, but few who put it in explicitly racial tones. And regardless of course, other soldiers in Spain is a far cry from 'people back in the USA'.

The issue surrounding Law is only further enhanced by the circumstances of his death, which I expand on here, and the unsettled - and likely never to be - debate over whether he was shot in the front, or in the back. But while the debates over his competency as a commander, or his bravery in the field, are tied up in discussion of race within the Abe Lincolns, they were limited to Spain itself.

Sources

Cecil D. Eby. Comrades and Commissars: The Lincoln Battalion in the Spanish Civil War. Penn State Press, 2007.

Peter Carroll, The Odyssey of the Abraham Lincoln Brigade. Stanford University Press, 1994.

Americans earn high posts with loyalists. 1937. Afro-American (1893-1988), Oct 16, 1937.

WILL OBSERVE OLIVER LAW'S DEATH, JULY 10: SPANISH WAR VETERANS TO HOLD SERVICE FOR LEADER. 1938. The Chicago Defender (National edition) (1921-1967), Jul 09, 1938

Wireless to THE NEW YORK TIMES. 1937. American,BattalionHead, killed in madrid action. New York Times (1923-Current file), Jul 14, 1937.

Cunard, Nancy. 1938. Tells how commander died in war: Few volunteers in spanish war now left. New Journal and Guide (1916-2003)1938.

Abe lincoln battalion leader killed in spain. 1937. The Pittsburgh Courier (1911-1950), Jul 24, 1937.

Huge air fleet rains bombs on insurgent lines: Loyalist flyers drone over madrid front as miaja continues thrust west rebel airmen' counter 74 craft engage in battle, with 12 shot to earth. 1937. New York Herald Tribune (1926-1962), Jul 13, 1937.

MADRID CLAIMS NEW GAINS IN DRIVE TO WEST: REBELS DEFEATED BY SURPRISE ATTACK NEAR EXTREMADURA ROAD FASCISTS REPORT OFFENSIVE BROKEN AVIATION FORCES OF BOTH SIDES ACTIVE, BOMBING OPPOSING LINES. 1937. The Sun (1837-1995), Jul 11, 1937.