TLDR: Harmony can work on 106~ cell samples but has less versatility then methods like BATMAN. BATMAN is useful if you need your data for differential gene expression and single-cell eQTL and can work on ~200 cell data.
Well their certainly is a minimum which comes down to several factors. As many of these rely on algorithms like KNN and other clustering algorithms for integration the threshold is dependent on the quality of the data and the algorithms abilities to predict reasonable clusters from that data. As well as the downstream uses of the data...
BATMAN highest performance on simulated set 200 cells and 1000 cells
With 200 cells from one batch and and 1000 in the other only BATMAN succeeded in clustering the high dimensionality simulated data well. If you need your data for differential gene expression and single-cell eQTL, BATMAN seems to be the correct choice, for small sets. "The downside of these methods is that they operate in latent space, which limits their interpretability and use in downstream analyses such as differential gene expression and single-cell eQTL analyses."
"A shows that the two original datasets become better mixed with each other. BATMAN has the highest performance not only when considering the top two principal components but also when considering more top principal components. It is the only method that manages to efficiently maintain a high iLISI score in a larger number of dimensions."
Harmony is best if... differential gene expression and single-cell eQTL is not necessary
"Moreover, we show that Harmony requires dramatically fewer computational resources. It is the only available algorithm that makes the integration of ~106 cells feasible on a personal computer."
Computing time benchmarks-section
It seems as the runtime of these algorithms is less than 2 minutes, using a set with less than ~2000 cells will not benefit your computational time much.
"When processing a small data set of ~ 2000 cells, all four methods took less than 2 min."
"To obtain such datasets, we downsampled the MCA and TM datasets to obtain a total of 9 sets of data containing between ~ 2000 and ~ 140,000 cells, while the number of highly variable genes (HVGs) was controlled in a range from ~ 2000 to ~ 3000 (Table S1)."