I came up with a possible solution, so I will attempt to answer my own question. I would like some critical feedback from the community though.
I know the two phenomenon are related, so I make the assumption that I can calibrate one scale to the other scale. I will then compare agreement between the predicted values from one method to the experimental values of the other method. This method still cannot find bias in means (as @Jeremy pointed out, this isn't meaningful in this context), but it still might allow a comparison of the 95% limits.
Some code (in R) to compare:
library(ggplot2)
set.seed(2063) #Dr. Cochrane
bland <- function(x, y, titl=''){
gg.data <- data.frame(x=x, y=y, avg=(x+y)/2, diff=(x-y))
g <- ggplot(gg.data, aes(x=avg, y=diff)) + geom_point(size=4) + theme_bw()
g <- g + theme(text=element_text(size=24), axis.text=element_text(colour='black'))
g <- g + labs(x='Average', y='Difference') + ggtitle(titl)
g <- g + geom_hline(yintercept=mean(gg.data$diff), colour='chocolate', size=1)
g <- g + geom_hline(yintercept=mean(gg.data$diff) + 1.96*sd(gg.data$diff), colour='dodgerblue3', size=1,
linetype='dashed')
g <- g + geom_hline(yintercept=mean(gg.data$diff) - 1.96*sd(gg.data$diff), colour='dodgerblue3', size=1,
linetype='dashed')
plot(g)
}
#Make some data
x <- seq(1,40)
y <- 2*x + rnorm(n=length(x), mean=0, sd=10)
qplot(x,y)
lm.data <- data.frame(x=x, y=y)
lm(data=lm.data, y~x)
#Bland-Altman of raw data
bland(x,y,'Raw Data')
#Bland-Altman of calibrated data
orig.df <- data.frame(x=x)
y.p <- predict(lm(data=lm.data, y~x), newdata=orig.df)
bland(y.p,y, 'Calib Data')
qplot(y.p,y)
If I try to directly compare $x$ and $y$, as expected, I get what would be very poor agreement:

However, if I "calibrate" the $x$ values to the $y$ scale using a linear model, agreement appears much better:

Some key thoughts:
- I don't have to use a linear model. Any model that calibrates one scale to another would do nicely.
- This is functionally equivalent to plotting the model residuals against the mean of $y$ and the $\hat{y}$ value. This is my biggest concern. I want to compare agreement between methods, but I could be simply evaluating the quality of the model. My current thinking is that these two are equivalent.
- Given #2, by comparing the residuals of the model as a measure of agreement, the value of my comparison rests strongly on the assumption that the model used to calibrate is correct.
To bring it all together, if I have selected a reasonable model (#1) to calibrate one scale to another (#3), then I can reasonably compare the residuals of that model (#2) as a measure of agreement. In the 2nd example graph above, I would interpret this as 95% of all deviations are within ~20 points on the $y$ scale. I can then evaluate if these limits are reasonable for the two methods I'm trying to study.
As I said above, criticisms are welcome.