DeLong’s test may not be suitable for nested model comparisons (e.g. https://pubmed.ncbi.nlm.nih.gov/22415937/). You may consider R^2 based test (https://www.sciencedirect.com/science/article/pii/S0002929723000046), which is available in CRAN (https://cran.r-project.org/web/packages/r2redux/index.html).
For your model comparisons, there are two ways.
Model_A : y = pred1 + pred2 + e
Model_B : y = pred1 + pred2 + pred3 + e
mod = lm (y ~ pred1 + pred2 + pred3)
merged_predictor = cbind(pred1, pred2) %*% mod$coefficients[2:3]
Then, the comparison can be equivalently expressed as
Model_A : y = merged_predictor + e
Model_B : y = merged_predictor + pred3 + e
Then, you can use r2redux as
r2_diff(dat,c(1,2),1,nv) (please see manual in https://cran.r-project.org/web/packages/r2redux/index.html).
Note that whether using lm or glm with logit link wouldn’t be really matter, i.e. the result would be very similar.
Secondly, you can use preadjusted y as
Model_B : y* = pred3 + e
where y* is adjusted y for pred1 and pred2, i.e. y* = mod2$residuals with mod2 = lm(y ~ pred1 + pred2).
Then, you can use r2redux as
r2_var(dat,1,nv) with properly specified inputs (please see manual in https://cran.r-project.org/web/packages/r2redux/index.html).
Additional note. There is one-to-one mapping between R^2 and AUC (https://cran.r-project.org/web/packages/R2ROC/index.html), which can also be used in such model comparisons. Please see https://www.biorxiv.org/content/10.1101/2023.08.01.551571v1.full.