[test] check miner doesn't select 0fee transactions

diff --git a/test/functional/mining_basic.py b/test/functional/mining_basic.py
index a4e482f6..6707f4b0 100755
--- a/test/functional/mining_basic.py
+++ b/test/functional/mining_basic.py
@@ -39,6 +39,7 @@ from test_framework.p2p import P2PDataStore
 from test_framework.test_framework import BitcoinTestFramework
 from test_framework.util import (
     assert_equal,
+    assert_greater_than,
     assert_greater_than_or_equal,
     assert_raises_rpc_error,
     get_fee,
@@ -143,7 +144,7 @@ class MiningTest(BitcoinTestFramework):
         node = self.nodes[0]
 
         # test default (no parameter), zero and a bunch of arbitrary blockmintxfee rates [sat/kvB]
-        for blockmintxfee_sat_kvb in (DEFAULT_BLOCK_MIN_TX_FEE, 0, 50, 100, 500, 2500, 5000, 21000, 333333, 2500000):
+        for blockmintxfee_sat_kvb in (DEFAULT_BLOCK_MIN_TX_FEE, 0, 1, 5, 10, 50, 100, 500, 2500, 5000, 21000, 333333, 2500000):
             blockmintxfee_btc_kvb = blockmintxfee_sat_kvb / Decimal(COIN)
             if blockmintxfee_sat_kvb == DEFAULT_BLOCK_MIN_TX_FEE:
                 self.log.info(f"-> Default -blockmintxfee setting ({blockmintxfee_sat_kvb} sat/kvB)...")
@@ -154,19 +155,27 @@ class MiningTest(BitcoinTestFramework):
                 self.wallet.rescan_utxos()  # to avoid spending outputs of txs that are not in mempool anymore after restart
 
             # submit one tx with exactly the blockmintxfee rate, and one slightly below
-            tx_with_min_feerate = self.wallet.send_self_transfer(from_node=node, fee_rate=blockmintxfee_btc_kvb)
+            tx_with_min_feerate = self.wallet.send_self_transfer(from_node=node, fee_rate=blockmintxfee_btc_kvb, confirmed_only=True)
             assert_equal(tx_with_min_feerate["fee"], get_fee(tx_with_min_feerate["tx"].get_vsize(), blockmintxfee_btc_kvb))
-            if blockmintxfee_btc_kvb > 0:
+            if blockmintxfee_sat_kvb > 5:
                 lowerfee_btc_kvb = blockmintxfee_btc_kvb - Decimal(10)/COIN  # 0.01 sat/vbyte lower
-                tx_below_min_feerate = self.wallet.send_self_transfer(from_node=node, fee_rate=lowerfee_btc_kvb)
+                tx_below_min_feerate = self.wallet.send_self_transfer(from_node=node, fee_rate=lowerfee_btc_kvb, confirmed_only=True)
                 assert_equal(tx_below_min_feerate["fee"], get_fee(tx_below_min_feerate["tx"].get_vsize(), lowerfee_btc_kvb))
             else:  # go below zero fee by using modified fees
-                tx_below_min_feerate = self.wallet.send_self_transfer(from_node=node, fee_rate=blockmintxfee_btc_kvb)
+                tx_below_min_feerate = self.wallet.send_self_transfer(from_node=node, fee_rate=blockmintxfee_btc_kvb, confirmed_only=True)
                 node.prioritisetransaction(tx_below_min_feerate["txid"], 0, -1)
 
             # check that tx below specified fee-rate is neither in template nor in the actual block
             block_template = node.getblocktemplate(NORMAL_GBT_REQUEST_PARAMS)
             block_template_txids = [tx['txid'] for tx in block_template['transactions']]
+
+            # Unless blockmintxfee is 0, the template shouldn't contain free transactions.
+            # Note that the real block assembler uses package feerates, but we didn't create dependent transactions so it's ok to use base feerate.
+            if blockmintxfee_btc_kvb > 0:
+                for txid in block_template_txids:
+                    tx = node.getmempoolentry(txid)
+                    assert_greater_than(tx['fees']['base'], 0)
+
             self.generate(self.wallet, 1, sync_fun=self.no_op)
             block = node.getblock(node.getbestblockhash(), verbosity=2)
             block_txids = [tx['txid'] for tx in block['tx']]