A "typical" rate of twist for a picket rifle cut with the gain twist is usually from around 1/48" to 1/24". It's not really necessary to measure the angles because picket rifles use a bullet of less than 1 1/2 calibers in length. You can if you wish to know exactly, but any disparity between bullet length and twist rate can be countered by tailoring the powder charge.
Early picket bullets had rounded bases along with pointed noses, but these fell out of favor after a short time among target shooters when flat base bullets began to be used. The nose of your guide starter, if original, would have been cut with the cherry used to make the mould, and would keep from deforming the bullet during loading.
Many different types of bullets were tried but the best ones were relieved in diameter just below the ogive to allow room for the cloth patch to "bunch up" when loaded. This, coupled with the flat base sought to eliminate the tipping that occasionally caused fliers with the old style bullets, and worked pretty well. Others used this style bullet as well, and began doing so in the late 1840's.
Some rifles were designed for target use and hunting. These are sometimes called combination rifles. They were made to be used with both a round ball and picket bullet, with the muzzle of the barrel turned round. The guide starter was slid over the round part, where the cloth patched bullet could be loaded. Rifles with Patent, or loading (false) muzzles were usually target-only affairs. They were expensive, and were not made to be used with a round ball.
A .390 caliber picket bullet of proper length (.625") would probably weigh about 175 grains, when made of pure lead. Paper patching, Chase patching, grease grooved bullets all came along after the slug rifle was in use. Slug rifles used bullets of over 1 1/2 calibers in length.
The photo shows a bullet I use in a .38 caliber picket rifle made in Ohio around 1850. There are three of them, so you can see how it looks from different angles. This bullet weighs 165 grains, of pure lead. The large projectile is a paper-patched slug bullet, for a rifle built in 1885. More than 1 1/2 calibers in length.
Hope this helps. I've spent almost 30 years shooting original slug and picket rifles, and try to pass along what I have learned from others in order to get more rifles on the line.