You can call a click command function from regular code by reconstructing the command line from parameters.  Using your example it could look somthing like this:
call_click_command(app, width, [... other arguments ...])
Code:
def call_click_command(cmd, *args, **kwargs):
    """ Wrapper to call a click command
    :param cmd: click cli command function to call 
    :param args: arguments to pass to the function 
    :param kwargs: keywrod arguments to pass to the function 
    :return: None 
    """
    # Get positional arguments from args
    arg_values = {c.name: a for a, c in zip(args, cmd.params)}
    args_needed = {c.name: c for c in cmd.params
                   if c.name not in arg_values}
    # build and check opts list from kwargs
    opts = {a.name: a for a in cmd.params if isinstance(a, click.Option)}
    for name in kwargs:
        if name in opts:
            arg_values[name] = kwargs[name]
        else:
            if name in args_needed:
                arg_values[name] = kwargs[name]
                del args_needed[name]
            else:
                raise click.BadParameter(
                    "Unknown keyword argument '{}'".format(name))
    # check positional arguments list
    for arg in (a for a in cmd.params if isinstance(a, click.Argument)):
        if arg.name not in arg_values:
            raise click.BadParameter("Missing required positional"
                                     "parameter '{}'".format(arg.name))
    # build parameter lists
    opts_list = sum(
        [[o.opts[0], str(arg_values[n])] for n, o in opts.items()], [])
    args_list = [str(v) for n, v in arg_values.items() if n not in opts]
    # call the command
    cmd(opts_list + args_list)
How does this work?
This works because click is a well designed OO framework. The @click.Command object can be introspected to determine what parameters it is expecting.  Then a command line can be constructed that will look like the command line that click is expecting.
Test Code:
import click
@click.command()
@click.option('-w', '--width', type=int, default=0)
@click.option('--option2')
@click.argument('argument')
def app(width, option2, argument):
    click.echo("params: {} {} {}".format(width, option2, argument))
    assert width == 3
    assert option2 == '4'
    assert argument == 'arg'
width = 3
option2 = 4
argument = 'arg'
if __name__ == "__main__":
    commands = (
        (width, option2, argument, {}),
        (width, option2, dict(argument=argument)),
        (width, dict(option2=option2, argument=argument)),
        (dict(width=width, option2=option2, argument=argument),),
    )
    import sys, time
    time.sleep(1)
    print('Click Version: {}'.format(click.__version__))
    print('Python Version: {}'.format(sys.version))
    for cmd in commands:
        try:
            time.sleep(0.1)
            print('-----------')
            print('> {}'.format(cmd))
            time.sleep(0.1)
            call_click_command(app, *cmd[:-1], **cmd[-1])
        except BaseException as exc:
            if str(exc) != '0' and \
                    not isinstance(exc, (click.ClickException, SystemExit)):
                raise
Test Results:
Click Version: 6.7
Python Version: 3.6.3 (v3.6.3:2c5fed8, Oct  3 2017, 18:11:49) [MSC v.1900 64 bit (AMD64)]
-----------
> (3, 4, 'arg', {})
params: 3 4 arg
-----------
> (3, 4, {'argument': 'arg'})
params: 3 4 arg
-----------
> (3, {'option2': 4, 'argument': 'arg'})
params: 3 4 arg
-----------
> ({'width': 3, 'option2': 4, 'argument': 'arg'},)
params: 3 4 arg