Light GBM Regressor, L1 & L2 Regularization and Feature Importances
With regularization, LightGBM "shrinks" features which are not "helpful". So it is in fact normal, that feature importance is quite different with/without regularization. You don't need to exclude any features since the purpose of shrinking is to use features according to their importance (this happens automatically). 

In your case the top two features seem to have good explanatory power, so that they are used as "most important" features. Other features are less important and are therefore "shrunken" by the model. 

You may also find that different features pop up as top of the list (the list may look different in general) when you run the model multiple times. This is because (if you don't fix a seed), the model will take different pathes to obtain a best fit (so the whole thing is not deterministic).

Overall you should get a better fit with regularization (otherwise there is little need for it). 

I wonder if it makes sense to use both (l1 and l2)!? L1 (aka reg_alpha) can shrink features to zero while l2 (aka reg_lambda) does not. I usually only use one of the parameters. Unfortunately, the documentation does not provide too much details here.