Is it alright to split a GridSearchCV?
First my understanding of your problem. You want to find the best hyperparameters for a Random Forest. 

For that, you want to first adjust n_estimators parameter and then the rest of parameters in different runs.

Before answering to your question, you will only want to do a thorough search of hyperparameters when you want to have an improvement of around 1%. So it will be a small improvement. If you want to improve your model, probably feature engineering or data engineering will give you a better improvement. Or even a different algorithm.

You can print the results of your GridsearchCV with

pd.DataFrame(clf.cv_results_)


The answer to your question:

No you shouldn't run a GridsearchCV in different runs you have to explore the whole parameters if you want to find the global minima. A small change in one parameter can affect other. At the end you are exploring a search space.