Chris
The rifle was described as a Winchester with several custom features, one of which was; double set triggers with the front trigger being very small in diameter and of a light let-off. The barrel was described as full octagon with matting on top three flats, no maker specified. In the equipment list for that match, his rifle was listed only as being Winchester. Some other rifles were listed with barrel maker such as Pope, Zischang, etc. so I assume (but don't know) Strecker's was factory. 
After posting the above paragraph of my response, I was looking for a photo of Strecker and found one from 1901 clearly showing the muzzle of Strecker's rifle and the 5 holes in it are a dead giveaway that his Winchester was a muzzle loader. NOW, I do not know if that is the same barrel that was on that rifle in 1895 though as he was shooting fixed ammo at the match in '95. I'm fairly sure it is the same rifle, as he used that Winchester throughout the remainder of his shooting career from the time he acquired it. (IMU, that rifle is still in the possession of the Strecker family)

While in the following comments, the arms in question are different I think the  fouling issue is the same.  
  In the Revolutionary war period, Patrick Ferguson developed a 'breech-loading" single-shot flintlock using a fast-twist trigger-guard mounted vertical breech plug.   He demonstrated it extensively to the British military establishment.  Said demos included very long strings of fire without failures---including some in a rainstorm.  
While the military  was amazed they failed to adopt it on a wide basis.  A company of light infantry was raised experimentally, and equipped as scout-snipers but upon arrival they were never issued their single-shot breech-loading flintlocks and the unit was dispersed as replacements to other LI companies.    I have read extensively on this and have never determined whether it was simply "establishment intransigence" or whether Ferguson realized the environmental conditions mitigated against the rifles use.  Ferguson was later killed at Kings Mountain where he was leading the Tory insurgency forces.

In the late 1950's and 60's an old friend of mine Matt Switlick of Monroe MI recreated one of those rifles and used it to put on public exhibitions.  I shot it a number of times.   It did bore foul though the tight fitting ball being breech-loaded tended to keep the bore clear until it got too hot and dried the fouling between shots.
It's criticall problem was with the powder-fouling developing on the breech-plug screw and disabling the rifle; unless used as a standard muzzle loader using balls with more windage.

Matt's conclusion was that the MUCH lower humidity in the colonies permitted the fouling to build up and dry out putting the mechanism out of action.    Back in England, especially during the "rainstorm trials",  the relative humidity kept the fouling wet and soft. Some Ferguson rifles were later sent to India; under private purchase, by the East India Company's militia.  There,  they apparently worked well in high humidity ares.   Evidence indicates that a "special lubricant" was later developed to keep the Breech threads functioning in heavy combat situations. 

I would suspect that their descendant Black Powder cartridge rifles worked well in the higher relative humidity coastal conditions----especially if "blow tubes" or even if just blowing down the bore, was permitted.     I think the key to the long firing strings with BP was the ambient humidity at the event venues.

The optional solution is going to depend on a lot of factors. Guy that wins the match will figure it out. Single method won't work as well as one that suits the match rifle and conditions.

Boats

RSW, by pure coincidence I was flipping through the SSE Classic Compendium Vol #1 and noticed an article titled "Fast Shooting with Winchester Rifles".  It tells of an old match (circa 1889) for $500 to make the most offhand bullseyes in 3 hours.

Hazard Fg was used in likely .38-55 rifles with Zettler Bros barrels.  Rifles were switched out every 15-20 shots to cool them down and wipe out the barrels.

One shooter fired 334 shots in 60 minutes, 247 shots in the black.

I thought you might find that interesting, and somewhat relevant.

Chris.

I will read the article again to find more details, but the distance was 200 yards and I believe that the target was the German 25 ring.   They did talk about the barrels getting too hot.  I've never experienced that unless shooting in the hot sun, but then I've never tried shooting a round every 10 seconds

Chris.