Use this layout as a custom view inside a TabLayout.Tab to keep setText and setIcon working as usual

custom_font_tab_layout.xml

<?xml version="1.0" encoding="utf-8"?>
<LinearLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:orientation="vertical"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

    <!-- As long as BOTH these ids exist within your layout file
         TabLayout will automatically pull them in. 
         Then you can still use setText and setIcon as usual-->
    <ImageView
        android:id="@android:id/icon"
        android:layout_width="24dp"
        android:layout_height="24dp"
        android:scaleType="centerInside"/>


    <com.myapplication.CustomFontTextView
        android:id="@android:id/text1"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:gravity="center"
        app:fontName="Gotham-Medium.ttf"/>

</LinearLayout>

CustomFontTabLayout.java

public class CustomFontTabLayout extends TabLayout {

    public CustomFontTabLayout(Context context) {
        super(context);
    }

    public CustomFontTabLayout(Context context, AttributeSet attrs) {
        super(context, attrs);
    }

    public CustomFontTabLayout(Context context, AttributeSet attrs, int defStyleAttr) {
        super(context, attrs, defStyleAttr);
    }

    @NonNull
    @Override
    public Tab newTab() {
        Tab tab = super.newTab();
        tab.setCustomView(R.layout.custom_font_tab_layout);
        if(getTabTextColors() != null){
            //apply our color treatment to tabs when we create them.
            setTabTextColor(tab, getTabTextColors());
        }
        return tab;
    }

    @Override
    public void setTabTextColors(ColorStateList textColor) {
        super.setTabTextColors(textColor);

        //Custom views don't get the color treatment set it here whenever setTabTextColors gets called
        for(int tabIter = 0; tabIter < this.getTabCount(); tabIter++){
            setTabTextColor(getTabAt(tabIter), textColor);
        }
    }

    private void setTabTextColor(Tab tab, ColorStateList textColor){
        TextView textView = (TextView) tab.getCustomView().findViewById(android.R.id.text1);
        if(textView != null){
          textView.setTextColor(textColor);
        }
    }
}