To fix this, I ended up having to abandon the RadioGroup and RadioButtons, and I went with a simple Linear Layout with Buttons. It still made for relatively simple code, and it just involved changing up the Java code a bit to work with normal Buttons rather than RadioButtons. Here is the changed layout, if interested:
<LinearLayout android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:id="@+id/tab_radio_group"
android:layout_alignParentBottom="true"
android:orientation="horizontal"
android:background="@drawable/navigation_base"
android:paddingTop="15dp">
<Button
android:id="@+id/button_scan"
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"
android:background="@android:color/transparent"
android:drawableTop="@drawable/scan"/>
<Button
android:id="@+id/button_view_order"
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"
android:background="@android:color/transparent"
android:drawableTop="@drawable/view_order"/>
<Button
android:id="@+id/button_complete_order"
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"
android:background="@android:color/transparent"
android:drawableTop="@drawable/complete_order"/>
<Button
android:id="@+id/button_settings"
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"
android:background="@android:color/transparent"
android:drawableTop="@drawable/settings"/>
</LinearLayout>